Theorem csbief 3142

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 3141	. 2 |- ( A e. _V -> [_ A / x ]_ B = C )
6	1, 5	ax-mp 5	1 |- [_ A / x ]_ B = C

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   F/_wnfc 2336   _Vcvv 2773   [_csb 3097

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003  df-csb 3098

This theorem is referenced by:  csbie  3143  csbing  3384  csbopabg  4133  pofun  4372  csbima12g  5057  csbiotag  5278  csbriotag  5930  csbov123g  6001  eqerlem  6669  zsumdc  11780