Theorem csbiegf 3124

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 1460	. 2 |- A. x ( x = A -> B = C )
3		csbiegf.1	. . 3 |- ( A e. V -> F/_ x C )
4		csbiebt 3120	. . 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 1362   = wceq 1364   e. wcel 2164   F/_wnfc 2323   [_csb 3080

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-csb 3081

This theorem is referenced by:  csbief  3125  sbcco3g  3138  csbco3g  3139  fmptcof  5725  fmpoco  6269  iseqf1olemjpcl  10579  iseqf1olemqpcl  10580  iseqf1olemfvp  10581  seq3f1olemqsum  10584  sumsnf  11552  prodsnf  11735  pcmpt  12481