Theorem sbciegf 2483

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 1664	. 2 |- (A e. B -> A.x(ps -> A.xps))
3		sbciegf.2	. . . 4 |- (x = A -> (ph <-> ps))
4	3	ax-gen 1305	. . 3 |- A.x(x = A -> (ph <-> ps))
5		sbciegft 2482	. . . 4 |- ((A e. B /\ A.x(ps -> A.xps) /\ A.x(x = A -> (ph <-> ps))) -> ([A / x]ph <-> ps))
6	5	3exp 1066	. . 3 |- (A e. B -> (A.x(ps -> A.xps) -> (A.x(x = A -> (ph <-> ps)) -> ([A / x]ph <-> ps))))
7	4, 6	mpii 56	. 2 |- (A e. B -> (A.x(ps -> A.xps) -> ([A / x]ph <-> ps)))
8	2, 7	mpd 29	1 |- (A e. B -> ([A / x]ph <-> ps))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534

This theorem is referenced by:  sbcieg 2484  sbcbrgOLD 3391  bnj1464 13143  cbicplem 14508  cbvsbcOLD 15355  ac6gf 15749  fveqsb 16431

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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