Theorem csbief 3129

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 3128	. 2 |- ( A e. _V -> [_ A / x ]_ B = C )
6	1, 5	ax-mp 5	1 |- [_ A / x ]_ B = C

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   F/_wnfc 2326   _Vcvv 2763   [_csb 3084

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990  df-csb 3085

This theorem is referenced by:  csbie  3130  csbing  3371  csbopabg  4112  pofun  4348  csbima12g  5031  csbiotag  5252  csbriotag  5893  csbov123g  5964  eqerlem  6632  zsumdc  11566