Theorem hbsbcgd 1981

Description: Deduction version of hbsbcg 1948.

Hypotheses

Ref	Expression
hbsbcgd.1	|- (ph -> A.xph)
hbsbcgd.2	|- (ph -> A.yph)
hbsbcgd.3	|- (ph -> (z e. A -> A.x z e. A))
hbsbcgd.4	|- (ph -> (ps -> A.xps))

Assertion

Ref	Expression
hbsbcgd	|- ((ph /\ A e. C) -> ([A / y]ps -> A.x[A / y]ps))

Distinct variable groups:   z,A   ph,z   x,z

Proof of Theorem hbsbcgd

Step	Hyp	Ref	Expression
1		ax-4 972	. . . . . . . . 9 |- (A.x z e. A -> z e. A)
2		hbsbcgd.3	. . . . . . . . 9 |- (ph -> (z e. A -> A.x z e. A))
3	1, 2	impbid2 517	. . . . . . . 8 |- (ph -> (A.x z e. A <-> z e. A))
4	3	abbidv 1575	. . . . . . 7 |- (ph -> {z | A.x z e. A} = {z | z e. A})
5		eleq1 1532	. . . . . . . . 9 |- (z = w -> (z e. A <-> w e. A))
6	5	albidv 1277	. . . . . . . 8 |- (z = w -> (A.x z e. A <-> A.x w e. A))
7	6	cbvabv 1906	. . . . . . 7 |- {z | A.x z e. A} = {w | A.x w e. A}
8		abid2 1578	. . . . . . 7 |- {z | z e. A} = A
9	4, 7, 8	3eqtr3g 1528	. . . . . 6 |- (ph -> {w | A.x w e. A} = A)
10	9	eleq1d 1538	. . . . 5 |- (ph -> ({w | A.x w e. A} e. V <-> A e. V))
11	10	biimpar 417	. . . 4 |- ((ph /\ A e. V) -> {w | A.x w e. A} e. V)
12		hba1 1002	. . . . . 6 |- (A.x w e. A -> A.xA.x w e. A)
13	12	hbab 1466	. . . . 5 |- (z e. {w | A.x w e. A} -> A.x z e. {w | A.x w e. A})
14		hba1 1002	. . . . 5 |- (A.xps -> A.xA.xps)
15	13, 14	hbsbcg 1948	. . . 4 |- ({w | A.x w e. A} e. V -> ([{w | A.x w e. A} / y]A.xps -> A.x[{w | A.x w e. A} / y]A.xps))
16	11, 15	syl 10	. . 3 |- ((ph /\ A e. V) -> ([{w | A.x w e. A} / y]A.xps -> A.x[{w | A.x w e. A} / y]A.xps))
17	2	19.21aiv 1285	. . . . . 6 |- (ph -> A.z(z e. A -> A.x z e. A))
18		abidhb 1909	. . . . . 6 |- (A.z(z e. A -> A.x z e. A) -> {w | A.x w e. A} = A)
19		dfsbcq 1940	. . . . . 6 |- ({w | A.x w e. A} = A -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
20	17, 18, 19	3syl 20	. . . . 5 |- (ph -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
21	20	adantr 389	. . . 4 |- ((ph /\ A e. V) -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]A.xps))
22		hbsbcgd.2	. . . . 5 |- (ph -> A.yph)
23		ax-4 972	. . . . . 6 |- (A.xps -> ps)
24		hbsbcgd.4	. . . . . 6 |- (ph -> (ps -> A.xps))
25	23, 24	impbid2 517	. . . . 5 |- (ph -> (A.xps <-> ps))
26	22, 25	sbcbid 1973	. . . 4 |- ((ph /\ A e. V) -> ([A / y]A.xps <-> [A / y]ps))
27	21, 26	bitrd 527	. . 3 |- ((ph /\ A e. V) -> ([{w | A.x w e. A} / y]A.xps <-> [A / y]ps))
28		hbsbcgd.1	. . . . . . 7 |- (ph -> A.xph)
29	28	a1d 12	. . . . . 6 |- (ph -> (ph -> A.xph))
30		ax-17 970	. . . . . . . 8 |- (z e. V -> A.x z e. V)
31	30	a1i 8	. . . . . . 7 |- (ph -> (z e. V -> A.x z e. V))
32	28, 2, 31	hbeld 1911	. . . . . 6 |- (ph -> (A e. V -> A.x A e. V))
33	29, 32	hband 1110	. . . . 5 |- (ph -> ((ph /\ A e. V) -> A.x(ph /\ A e. V)))
34	33	anabsi5 495	. . . 4 |- ((ph /\ A e. V) -> A.x(ph /\ A e. V))
35	34, 27	albid 1103	. . 3 |- ((ph /\ A e. V) -> (A.x[{w | A.x w e. A} / y]A.xps <-> A.x[A / y]ps))
36	16, 27, 35	3imtr3d 541	. 2 |- ((ph /\ A e. V) -> ([A / y]ps -> A.x[A / y]ps))
37		elisset 1814	. 2 |- (A e. C -> A e. V)
38	36, 37	sylan2 451	1 |- ((ph /\ A e. C) -> ([A / y]ps -> A.x[A / y]ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  [wsbc 1169  {cab 1462  Vcvv 1808

This theorem is referenced by:  hbcsbgd 2025

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939

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