Theorem csbiegft 2573

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2575.)

Assertion

Ref	Expression
csbiegft	|- ((A e. D /\ A.xA.y(y e. C -> A.x y e. C) /\ A.x(x = A -> B = C)) -> [_A / x]_B = C)

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

Proof of Theorem csbiegft

Step	Hyp	Ref	Expression
1		sbciegft 2482	. . . 4 |- ((A e. D /\ A.x(z e. C -> A.x z e. C) /\ A.x(x = A -> (z e. B <-> z e. C))) -> ([A / x]z e. B <-> z e. C))
2		id 73	. . . 4 |- (A e. D -> A e. D)
3		visset 2295	. . . . . 6 |- z e. _V
4		eleq1 1957	. . . . . . 7 |- (y = z -> (y e. C <-> z e. C))
5	4	albidv 1656	. . . . . . 7 |- (y = z -> (A.x y e. C <-> A.x z e. C))
6	4, 5	imbi12d 688	. . . . . 6 |- (y = z -> ((y e. C -> A.x y e. C) <-> (z e. C -> A.x z e. C)))
7	3, 6	cla4v 2370	. . . . 5 |- (A.y(y e. C -> A.x y e. C) -> (z e. C -> A.x z e. C))
8	7	alimi 1338	. . . 4 |- (A.xA.y(y e. C -> A.x y e. C) -> A.x(z e. C -> A.x z e. C))
9		eleq2 1958	. . . . . 6 |- (B = C -> (z e. B <-> z e. C))
10	9	imim2i 11	. . . . 5 |- ((x = A -> B = C) -> (x = A -> (z e. B <-> z e. C)))
11	10	alimi 1338	. . . 4 |- (A.x(x = A -> B = C) -> A.x(x = A -> (z e. B <-> z e. C)))
12	1, 2, 8, 11	syl3an 1139	. . 3 |- ((A e. D /\ A.xA.y(y e. C -> A.x y e. C) /\ A.x(x = A -> B = C)) -> ([A / x]z e. B <-> z e. C))
13	12	abbi1dv 2010	. 2 |- ((A e. D /\ A.xA.y(y e. C -> A.x y e. C) /\ A.x(x = A -> B = C)) -> {z | [A / x]z e. B} = C)
14		df-csb 2541	. 2 |- [_A / x]_B = {z | [A / x]z e. B}
15	13, 14	syl5eq 1940	1 |- ((A e. D /\ A.xA.y(y e. C -> A.x y e. C) /\ A.x(x = A -> B = C)) -> [_A / x]_B = C)

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  [_csb 2540

This theorem is referenced by:  csbiegf 2575  csbnestglem 2580  csbnest1g 2582  csbco3g 2585  sbcco3g 2586

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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