Theorem csbief 3252

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 11	. . 3 |- ( A e. _V -> F/_ x C )
4		csbief.3	. . 3 |- ( x = A -> B = C )
5	3, 4	csbiegf 3251	. 2 |- ( A e. _V -> [_ A / x ]_ B = C )
6	1, 5	ax-mp 8	1 |- [_ A / x ]_ B = C

Syntax hints:   -> wi 4   = wceq 1649   e. wcel 1721   F/_wnfc 2527   _Vcvv 2916   [_csb 3211

This theorem is referenced by:  csbing  3508  csbifg  3727  csbopabg  4243  pofun  4479  csbima12g  5172  csbiotag  5406  csbovg  6071  csbriotag  6521  eqerlem  6896  fsum  12469  fsumcnv  12512  fsumshftm  12519  fsum0diag2  12521  ruclem1  12785  pcmpt  13216  odval  15127  psrass1lem  16397  iundisj2  19396  isibl  19610  dfitg  19614  dvfsumlem2  19864  mpfrcl  19892  fsumdvdsmul  20933  disjxpin  23981  iundisj2f  23983  iundisj2fi  24106  fprod  25220  fprodcnv  25260  bpolyval  25999  fphpd  26767  monotuz  26894  oddcomabszz  26897  fnwe2val  27014  fnwe2lem1  27015  mamufval  27311  csbafv12g  27868  csbaovg  27911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-sbc 3122  df-csb 3212