Theorem csbiegf 3048

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 1426	. 2 |- A. x ( x = A -> B = C )
3		csbiegf.1	. . 3 |- ( A e. V -> F/_ x C )
4		csbiebt 3044	. . 3 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
5	3, 4	mpdan 418	. 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 1330   = wceq 1332   e. wcel 1481   F/_wnfc 2269   [_csb 3007

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by:  csbief  3049  sbcco3g  3062  csbco3g  3063  fmptcof  5595  fmpoco  6121  iseqf1olemjpcl  10299  iseqf1olemqpcl  10300  iseqf1olemfvp  10301  seq3f1olemqsum  10304  sumsnf  11210