Theorem finds2 3153

Description: Principle of Finite Induction (inference schema) with implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136.

Hypotheses

Ref	Expression
finds2.1	|- (x = (/) -> (ph <-> ps))
finds2.2	|- (x = y -> (ph <-> ch))
finds2.3	|- (x = suc y -> (ph <-> th))
finds2.4	|- (ta -> ps)
finds2.5	|- (y e. om -> (ta -> (ch -> th)))

Assertion

Ref	Expression
finds2	|- (x e. om -> (ta -> ph))

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

Proof of Theorem finds2

Step	Hyp	Ref	Expression
1		finds2.4	. . . . 5 |- (ta -> ps)
2		0ex 2706	. . . . . 6 |- (/) e. V
3		finds2.1	. . . . . . 7 |- (x = (/) -> (ph <-> ps))
4	3	imbi2d 611	. . . . . 6 |- (x = (/) -> ((ta -> ph) <-> (ta -> ps)))
5	2, 4	elab 1893	. . . . 5 |- ((/) e. {x | (ta -> ph)} <-> (ta -> ps))
6	1, 5	mpbir 190	. . . 4 |- (/) e. {x | (ta -> ph)}
7		finds2.5	. . . . . . 7 |- (y e. om -> (ta -> (ch -> th)))
8	7	a2d 13	. . . . . 6 |- (y e. om -> ((ta -> ch) -> (ta -> th)))
9		visset 1809	. . . . . . 7 |- y e. V
10		finds2.2	. . . . . . . 8 |- (x = y -> (ph <-> ch))
11	10	imbi2d 611	. . . . . . 7 |- (x = y -> ((ta -> ph) <-> (ta -> ch)))
12	9, 11	elab 1893	. . . . . 6 |- (y e. {x | (ta -> ph)} <-> (ta -> ch))
13	9	sucex 3045	. . . . . . 7 |- suc y e. V
14		finds2.3	. . . . . . . 8 |- (x = suc y -> (ph <-> th))
15	14	imbi2d 611	. . . . . . 7 |- (x = suc y -> ((ta -> ph) <-> (ta -> th)))
16	13, 15	elab 1893	. . . . . 6 |- (suc y e. {x | (ta -> ph)} <-> (ta -> th))
17	8, 12, 16	3imtr4g 552	. . . . 5 |- (y e. om -> (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)}))
18	17	rgen 1695	. . . 4 |- A.y e. om (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)})
19		peano5 3148	. . . 4 |- (((/) e. {x | (ta -> ph)} /\ A.y e. om (y e. {x | (ta -> ph)} -> suc y e. {x | (ta -> ph)})) -> om (_ {x | (ta -> ph)})
20	6, 18, 19	mp2an 696	. . 3 |- om (_ {x | (ta -> ph)}
21	20	sseli 2061	. 2 |- (x e. om -> x e. {x | (ta -> ph)})
22		abid 1463	. 2 |- (x e. {x | (ta -> ph)} <-> (ta -> ph))
23	21, 22	sylib 198	1 |- (x e. om -> (ta -> ph))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  {cab 1461  A.wral 1642   (_ wss 2043  (/)c0 2276  suc csuc 2945  omcom 3126

This theorem is referenced by:  finds1 3154  omsmolem 4246  unblem2 4524  fiint 4540  trcl 4625  alephfplem3 4878

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127

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