Theorem csbiegf 3138

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 1473	. 2 |- A. x ( x = A -> B = C )
3		csbiegf.1	. . 3 |- ( A e. V -> F/_ x C )
4		csbiebt 3134	. . 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 1371   = wceq 1373   e. wcel 2177   F/_wnfc 2336   [_csb 3094

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3000  df-csb 3095

This theorem is referenced by:  csbief  3139  sbcco3g  3152  csbco3g  3153  fmptcof  5754  fmpoco  6309  iseqf1olemjpcl  10660  iseqf1olemqpcl  10661  iseqf1olemfvp  10662  seq3f1olemqsum  10665  sumsnf  11764  prodsnf  11947  pcmpt  12710