Theorem csbiegf 3181

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbiegf.1	|- ( A e. V -> F/_ x C )
csbiegf.2	|- ( x = A -> B = C )

Assertion

Ref	Expression
csbiegf	|- ( A e. V -> [_ A / x ]_ B = C )

Distinct variable groups:   x, A   x, V

Allowed substitution hints:   B( x)   C( x)

Proof of Theorem csbiegf

Step	Hyp	Ref	Expression
1		csbiegf.2	. . 3 |- ( x = A -> B = C )
2	1	ax-gen 1498	. 2 |- A. x ( x = A -> B = C )
3		csbiegf.1	. . 3 |- ( A e. V -> F/_ x C )
4		csbiebt 3177	. . 3 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
5	3, 4	mpdan 421	. 2 |- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
6	2, 5	mpbii 148	1 |- ( A e. V -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2203   F/_wnfc 2371   [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-sbc 3042  df-csb 3138

This theorem is referenced by:  csbief  3182  sbcco3g  3195  csbco3g  3196  fmptcof  5843  fmpoco  6411  iseqf1olemjpcl  10869  iseqf1olemqpcl  10870  iseqf1olemfvp  10871  seq3f1olemqsum  10874  sumsnf  12091  prodsnf  12274  pcmpt  13037