Theorem csbiegf 2034

Description: Conversion of implicit substitution to explicit substitution into a class.

Hypotheses

Ref	Expression
csbiegf.1	|- (A e. D -> (y e. C -> A.x y e. C))
csbiegf.2	|- (x = A -> B = C)

Assertion

Ref	Expression
csbiegf	|- (A e. D -> [_A / x]_B = C)

Distinct variable groups:   x,y,A   y,C   x,D,y

Proof of Theorem csbiegf

Step	Hyp	Ref	Expression
1		csbiegf.1	. . . 4 |- (A e. D -> (y e. C -> A.x y e. C))
2	1	19.21aivv 1289	. . 3 |- (A e. D -> A.xA.y(y e. C -> A.x y e. C))
3		csbiegf.2	. . . 4 |- (x = A -> B = C)
4	3	ax-gen 965	. . 3 |- A.x(x = A -> B = C)
5	2, 4	jctir 293	. 2 |- (A e. D -> (A.xA.y(y e. C -> A.x y e. C) /\ A.x(x = A -> B = C)))
6		csbiegft 2032	. . 3 |- ((A e. D /\ A.xA.y(y e. C -> A.x y e. C) /\ A.x(x = A -> B = C)) -> [_A / x]_B = C)
7	6	3expb 836	. 2 |- ((A e. D /\ (A.xA.y(y e. C -> A.x y e. C) /\ A.x(x = A -> B = C))) -> [_A / x]_B = C)
8	5, 7	mpdan 706	1 |- (A e. D -> [_A / x]_B = C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  [_csb 2004

This theorem is referenced by:  csbima12g 3419  csbfv12g 3748  csboprg 3992  csbnegg 5376  fsum1p 7019

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945  df-csb 2005

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