Theorem sbciegf 1931

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 1268	. 2 |- (A e. B -> A.x(ps -> A.xps))
3		sbciegf.2	. . . 4 |- (x = A -> (ph <-> ps))
4	3	ax-gen 955	. . 3 |- A.x(x = A -> (ph <-> ps))
5		sbciegft 1930	. . . 4 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph <-> ps))
6	5	3exp 829	. . 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 950   = wceq 1099   e. wcel 1105  [wsbc 1153

This theorem is referenced by:  sbcieg 1932  sbcbrg 2630

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913

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