Theorem csbief 3186

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbief.1	|- A e. _V
csbief.2	|- F/_ x C
csbief.3	|- ( x = A -> B = C )

Assertion

Ref	Expression
csbief	|- [_ A / x ]_ B = C

Distinct variable group:   x, A

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

Proof of Theorem csbief

Step	Hyp	Ref	Expression
1		csbief.1	. 2 |- A e. _V
2		csbief.2	. . . 4 |- F/_ x C
3	2	a1i 9	. . 3 |- ( A e. _V -> F/_ x C )
4		csbief.3	. . 3 |- ( x = A -> B = C )
5	3, 4	csbiegf 3185	. 2 |- ( A e. _V -> [_ A / x ]_ B = C )
6	1, 5	ax-mp 5	1 |- [_ A / x ]_ B = C

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   F/_wnfc 2373   _Vcvv 2815   [_csb 3141

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-sbc 3046  df-csb 3142

This theorem is referenced by:  csbie  3187  csbing  3432  csbopabg  4193  pofun  4438  csbima12g  5128  csbiotag  5350  csbriotag  6025  csbov123g  6097  eqerlem  6811  zsumdc  12095