Theorem hbsbcg 1947

Description: Bound-variable hypothesis builder for class substitution. (The proof was shortened by Eric Schmidt, 17-Jan-2007.)

Hypotheses

Ref	Expression
hbsbcg.1	|- (z e. A -> A.x z e. A)
hbsbcg.2	|- (ph -> A.xph)

Assertion

Ref	Expression
hbsbcg	|- (A e. B -> ([A / y]ph -> A.x[A / y]ph))

Distinct variable groups:   z,A   x,z

Proof of Theorem hbsbcg

Step	Hyp	Ref	Expression
1		dfsbcq 1939	. . 3 |- (w = A -> ([w / y]ph <-> [A / y]ph))
2		ax-17 969	. . . . 5 |- (z e. w -> A.x z e. w)
3		hbsbcg.1	. . . . 5 |- (z e. A -> A.x z e. A)
4	2, 3	hbeq 1562	. . . 4 |- (w = A -> A.x w = A)
5	4, 1	albid 1102	. . 3 |- (w = A -> (A.x[w / y]ph <-> A.x[A / y]ph))
6	1, 5	imbi12d 625	. 2 |- (w = A -> (([w / y]ph -> A.x[w / y]ph) <-> ([A / y]ph -> A.x[A / y]ph)))
7		hbsbcg.2	. . 3 |- (ph -> A.xph)
8	7	hbsb 1331	. 2 |- ([w / y]ph -> A.x[w / y]ph)
9	6, 8	vtoclg 1843	1 |- (A e. B -> ([A / y]ph -> A.x[A / y]ph))

Syntax hints:   -> wi 3  A.wal 952   = wceq 954   e. wcel 956  [wsbc 1168

This theorem is referenced by:  hbsbcgd 1980  hbcsbg 2022  ralxpf 3215  dfopab2 4103  dfoprab3 4104

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-12 966  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

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