Theorem sbciegf 1950

Description: Conversion of implicit substitution to explicit class substitution.

Hypotheses

Ref	Expression
sbciegf.1	|- (A e. B -> (ps -> A.xps))
sbciegf.2	|- (x = A -> (ph <-> ps))

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem sbciegf

Step	Hyp	Ref	Expression
1		sbciegf.1	. . 3 |- (A e. B -> (ps -> A.xps))
2	1	19.21aiv 1281	. 2 |- (A e. B -> A.x(ps -> A.xps))
3		sbciegf.2	. . . 4 |- (x = A -> (ph <-> ps))
4	3	ax-gen 960	. . 3 |- A.x(x = A -> (ph <-> ps))
5		sbciegft 1949	. . . 4 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph <-> ps))
6	5	3exp 830	. . 3 |- (A e. B -> (A.x(ps -> A.xps) -> (A.x(x = A -> (ph <-> ps)) -> ([A / x]ph <-> ps))))
7	4, 6	mpii 45	. 2 |- (A e. B -> (A.x(ps -> A.xps) -> ([A / x]ph <-> ps)))
8	2, 7	mpd 26	1 |- (A e. B -> ([A / x]ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   = wceq 953   e. wcel 955  [wsbc 1166

This theorem is referenced by:  sbcieg 1951  sbcbrg 2652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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