Theorem csbiegf 2002

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 1269	. . 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 955	. . 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 2000	. . 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 831	. 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 701	1 |- (A e. D -> [_A / x]_B = C)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 950   = wceq 1099   e. wcel 1105  [_csb 1972

This theorem is referenced by:  csbima12g 3364  csbfv12g 3681  csboprg 3925  csbnegg 5287  fsum1p 6908

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913  df-csb 1973

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