Theorem hbsbcg 1445

Description: Bound variable hypothesis builder for class substitution.

Hypothesis

Ref	Expression
hbsbcg.1	|- (y e. A -> A.x y e. A)

Assertion

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

Distinct variable group(s):   x,y   y,A   y,B

Proof of Theorem hbsbcg

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. . 3 |- (z = A -> ([z / x]ph <-> [A / x]ph))
2		ax-17 925	. . . . . 6 |- (y e. z -> A.x y e. z)
3		hbsbcg.1	. . . . . 6 |- (y e. A -> A.x y e. A)
4	2, 3	hbeq 1171	. . . . 5 |- (z = A -> A.x z = A)
5	4, 1	biald 782	. . . 4 |- (z = A -> (A.x[z / x]ph <-> A.x[A / x]ph))
6		hbs1 986	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
7	5, 6	syl5bi 183	. . 3 |- (z = A -> ([z / x]ph -> A.x[A / x]ph))
8	1, 7	sylbird 180	. 2 |- (z = A -> ([A / x]ph -> A.x[A / x]ph))
9	8	vtocleg 1390	1 |- (A e. B -> ([A / x]ph -> A.x[A / x]ph))

Syntax hints:   -> wi 2  A.wal 672   e. wel 803  [wsb 852   = wceq 1091   e. wcel 1092  [wsbc 1440

This theorem is referenced by:  hbsbc 1446

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-sbc 1441

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