Theorem csbiegf 2575

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 1665	. . 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 1305	. . 3 |- A.x(x = A -> B = C)
5	2, 4	jctir 317	. 2 |- (A e. D -> (A.xA.y(y e. C -> A.x y e. C) /\ A.x(x = A -> B = C)))
6		csbiegft 2573	. . 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 1068	. 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 768	1 |- (A e. D -> [_A / x]_B = C)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  [_csb 2540

This theorem is referenced by:  csbima12g 4276  csbfv12g 4699  csboprgOLD 4911  csbneggOLD 6521  fsum1p 8279  unirep 15664  fsumlt1 15831

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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