Theorem csbiegft 2032

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2034.)

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 1962	. . . 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 59	. . . 4 |- (A e. D -> A e. D)
3		visset 1816	. . . . . 6 |- z e. V
4		eleq1 1537	. . . . . . 7 |- (y = z -> (y e. C <-> z e. C))
5	4	albidv 1280	. . . . . . 7 |- (y = z -> (A.x y e. C <-> A.x z e. C))
6	4, 5	imbi12d 628	. . . . . 6 |- (y = z -> ((y e. C -> A.x y e. C) <-> (z e. C -> A.x z e. C)))
7	3, 6	cla4v 1871	. . . . 5 |- (A.y(y e. C -> A.x y e. C) -> (z e. C -> A.x z e. C))
8	7	19.20i 994	. . . 4 |- (A.xA.y(y e. C -> A.x y e. C) -> A.x(z e. C -> A.x z e. C))
9		eleq2 1538	. . . . . 6 |- (B = C -> (z e. B <-> z e. C))
10	9	imim2i 17	. . . . 5 |- ((x = A -> B = C) -> (x = A -> (z e. B <-> z e. C)))
11	10	19.20i 994	. . . 4 |- (A.x(x = A -> B = C) -> A.x(x = A -> (z e. B <-> z e. C)))
12	1, 2, 8, 11	syl3an 870	. . 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 1582	. 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 2005	. 2 |- [_A / x]_B = {z | [A / x]z e. B}
15	13, 14	syl5eq 1522	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 146   /\ w3a 777  A.wal 956   = wceq 958   e. wcel 960  [wsbc 1172  {cab 1466  [_csb 2004

This theorem is referenced by:  csbiegf 2034  csbnestglem 2038  csbnest1g 2040  csbco3g 2043  sbcco3g 2044

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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