Theorem csbiegf 2031

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 1287	. . 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 963	. . 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 2029	. . 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 834	. 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 704	1 |- (A e. D -> [_A / x]_B = C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  [_csb 2001

This theorem is referenced by:  csbima12g 3413  csbfv12g 3742  csboprg 3986  csbnegg 5364  fsum1p 7019

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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