Theorem nn1suc 5941

Description: If a statement holds for 1 and also holds for a successor, it holds for all natural numbers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor.

Hypotheses

Ref	Expression
nn1suc.1	|- (x = 1 -> (ph <-> ps))
nn1suc.3	|- (x = (y + 1) -> (ph <-> ch))
nn1suc.4	|- (x = A -> (ph <-> th))
nn1suc.5	|- ps
nn1suc.6	|- (y e. NN -> ch)

Assertion

Ref	Expression
nn1suc	|- (A e. NN -> th)

Distinct variable groups:   x,y,A   ps,x   ch,x   th,x   ph,y

Proof of Theorem nn1suc

Step	Hyp	Ref	Expression
1		dfsbcq 1946	. . 3 |- (z = 1 -> ([z / x](A e. NN -> ph) <-> [1 / x](A e. NN -> ph)))
2		sbequ 1231	. . 3 |- (z = y -> ([z / x](A e. NN -> ph) <-> [y / x](A e. NN -> ph)))
3		dfsbcq 1946	. . 3 |- (z = (y + 1) -> ([z / x](A e. NN -> ph) <-> [(y + 1) / x](A e. NN -> ph)))
4		dfsbcq 1946	. . . . . . 7 |- (z = A -> ([z / x]ph <-> [A / x]ph))
5		elex 1822	. . . . . . . . . 10 |- (A e. NN -> E.x x = A)
6		ax-17 973	. . . . . . . . . . . . 13 |- (z e. A -> A.x z e. A)
7	6	hbsbc1 1952	. . . . . . . . . . . 12 |- ((A e. NN -> [A / x]ph) -> A.x(A e. NN -> [A / x]ph))
8		ax-17 973	. . . . . . . . . . . 12 |- ((A e. NN -> th) -> A.x(A e. NN -> th))
9	7, 8	hbbi 1012	. . . . . . . . . . 11 |- (((A e. NN -> [A / x]ph) <-> (A e. NN -> th)) -> A.x((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
10		sbceq1a 1947	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> [A / x]ph))
11		nn1suc.4	. . . . . . . . . . . . 13 |- (x = A -> (ph <-> th))
12	10, 11	bitr3d 532	. . . . . . . . . . . 12 |- (x = A -> ([A / x]ph <-> th))
13	12	imbi2d 614	. . . . . . . . . . 11 |- (x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
14	9, 13	19.23ai 1066	. . . . . . . . . 10 |- (E.x x = A -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
15	5, 14	syl 10	. . . . . . . . 9 |- (A e. NN -> ((A e. NN -> [A / x]ph) <-> (A e. NN -> th)))
16	15	pm5.74rd 590	. . . . . . . 8 |- (A e. NN -> (A e. NN -> ([A / x]ph <-> th)))
17	16	pm2.43i 64	. . . . . . 7 |- (A e. NN -> ([A / x]ph <-> th))
18	4, 17	sylan9bbr 543	. . . . . 6 |- ((A e. NN /\ z = A) -> ([z / x]ph <-> th))
19	18	expcom 374	. . . . 5 |- (z = A -> (A e. NN -> ([z / x]ph <-> th)))
20	19	pm5.74d 587	. . . 4 |- (z = A -> ((A e. NN -> [z / x]ph) <-> (A e. NN -> th)))
21		ax-17 973	. . . . 5 |- (A e. NN -> A.x A e. NN)
22	21	sb19.21 1238	. . . 4 |- ([z / x](A e. NN -> ph) <-> (A e. NN -> [z / x]ph))
23	20, 22	syl5bb 534	. . 3 |- (z = A -> ([z / x](A e. NN -> ph) <-> (A e. NN -> th)))
24		1nn 5936	. . . . . . . 8 |- 1 e. NN
25	24	elisseti 1821	. . . . . . 7 |- 1 e. V
26	25	isseti 1818	. . . . . 6 |- E.x x = 1
27	25	hbsbc1v 1953	. . . . . . 7 |- ([1 / x]ph -> A.x[1 / x]ph)
28		nn1suc.5	. . . . . . . . 9 |- ps
29		nn1suc.1	. . . . . . . . 9 |- (x = 1 -> (ph <-> ps))
30	28, 29	mpbiri 194	. . . . . . . 8 |- (x = 1 -> ph)
31		sbceq1a 1947	. . . . . . . 8 |- (x = 1 -> (ph <-> [1 / x]ph))
32	30, 31	mpbid 195	. . . . . . 7 |- (x = 1 -> [1 / x]ph)
33	27, 32	19.23ai 1066	. . . . . 6 |- (E.x x = 1 -> [1 / x]ph)
34	26, 33	ax-mp 7	. . . . 5 |- [1 / x]ph
35	34	a1i 8	. . . 4 |- (A e. NN -> [1 / x]ph)
36	21	sbc19.21g 1990	. . . . 5 |- (1 e. V -> ([1 / x](A e. NN -> ph) <-> (A e. NN -> [1 / x]ph)))
37	25, 36	ax-mp 7	. . . 4 |- ([1 / x](A e. NN -> ph) <-> (A e. NN -> [1 / x]ph))
38	35, 37	mpbir 190	. . 3 |- [1 / x](A e. NN -> ph)
39		nn1suc.6	. . . . . . 7 |- (y e. NN -> ch)
40		oprex 3989	. . . . . . . . 9 |- (y + 1) e. V
41	40	isseti 1818	. . . . . . . 8 |- E.x x = (y + 1)
42		ax-17 973	. . . . . . . . . 10 |- (ch -> A.xch)
43	40	hbsbc1v 1953	. . . . . . . . . 10 |- ([(y + 1) / x]ph -> A.x[(y + 1) / x]ph)
44	42, 43	hbbi 1012	. . . . . . . . 9 |- ((ch <-> [(y + 1) / x]ph) -> A.x(ch <-> [(y + 1) / x]ph))
45		nn1suc.3	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> ch))
46		sbceq1a 1947	. . . . . . . . . 10 |- (x = (y + 1) -> (ph <-> [(y + 1) / x]ph))
47	45, 46	bitr3d 532	. . . . . . . . 9 |- (x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
48	44, 47	19.23ai 1066	. . . . . . . 8 |- (E.x x = (y + 1) -> (ch <-> [(y + 1) / x]ph))
49	41, 48	ax-mp 7	. . . . . . 7 |- (ch <-> [(y + 1) / x]ph)
50	39, 49	sylib 198	. . . . . 6 |- (y e. NN -> [(y + 1) / x]ph)
51	50	a1d 12	. . . . 5 |- (y e. NN -> (A e. NN -> [(y + 1) / x]ph))
52	21	sbc19.21g 1990	. . . . . 6 |- ((y + 1) e. V -> ([(y + 1) / x](A e. NN -> ph) <-> (A e. NN -> [(y + 1) / x]ph)))
53	40, 52	ax-mp 7	. . . . 5 |- ([(y + 1) / x](A e. NN -> ph) <-> (A e. NN -> [(y + 1) / x]ph))
54	51, 53	sylibr 200	. . . 4 |- (y e. NN -> [(y + 1) / x](A e. NN -> ph))
55	54	a1d 12	. . 3 |- (y e. NN -> ([y / x](A e. NN -> ph) -> [(y + 1) / x](A e. NN -> ph)))
56	1, 2, 3, 23, 38, 55	nnind 5939	. 2 |- (A e. NN -> (A e. NN -> th))
57	56	pm2.43i 64	1 |- (A e. NN -> th)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172  Vcvv 1814  (class class class)co 3969  1c1 5247   + caddc 5249  NNcn 5308

This theorem is referenced by:  nnleltp1t 5956  ruclem29 7539

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-sub 5368  df-neg 5370  df-n 5927

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