Theorem csbief 3178

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/_xC
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/_xC
3	2	a1i 10	. . 3 |- (A e. _V -> F/_xC)
4		csbief.3	. . 3 |- (x = A -> B = C)
5	3, 4	csbiegf 3177	. 2 |- (A e. _V -> [_A / x]_B = C)
6	1, 5	ax-mp 5	1 |- [_A / x]_B = C

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710   F/_wnfc 2477  _Vcvv 2860  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbing  3463  csbifg  3691  csbiotag  4372  csbopabg  4638  csbima12g  4956  csbovg  5553  eqerlem  5961