Theorem csbiegf 3043

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 1425	. 2 |- A. x ( x = A -> B = C )
3		csbiegf.1	. . 3 |- ( A e. V -> F/_ x C )
4		csbiebt 3039	. . 3 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
5	3, 4	mpdan 417	. 2 |- ( A e. V -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
6	2, 5	mpbii 147	1 |- ( A e. V -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480   F/_wnfc 2268   [_csb 3003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004

This theorem is referenced by:  csbief  3044  sbcco3g  3057  csbco3g  3058  fmptcof  5587  fmpoco  6113  iseqf1olemjpcl  10268  iseqf1olemqpcl  10269  iseqf1olemfvp  10270  seq3f1olemqsum  10273  sumsnf  11178