Theorem csbiegft 2029

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2031.)

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 1959	. . . 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 1813	. . . . . 6 |- z e. V
4		eleq1 1534	. . . . . . 7 |- (y = z -> (y e. C <-> z e. C))
5	4	albidv 1278	. . . . . . 7 |- (y = z -> (A.x y e. C <-> A.x z e. C))
6	4, 5	imbi12d 626	. . . . . 6 |- (y = z -> ((y e. C -> A.x y e. C) <-> (z e. C -> A.x z e. C)))
7	3, 6	cla4v 1868	. . . . 5 |- (A.y(y e. C -> A.x y e. C) -> (z e. C -> A.x z e. C))
8	7	19.20i 992	. . . 4 |- (A.xA.y(y e. C -> A.x y e. C) -> A.x(z e. C -> A.x z e. C))
9		eleq2 1535	. . . . . 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 992	. . . 4 |- (A.x(x = A -> B = C) -> A.x(x = A -> (z e. B <-> z e. C)))
12	1, 2, 8, 11	syl3an 868	. . 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 1579	. 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 2002	. 2 |- [_A / x]_B = {z | [A / x]z e. B}
15	13, 14	syl5eq 1519	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 775  A.wal 954   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463  [_csb 2001

This theorem is referenced by:  csbiegf 2031  csbnestglem 2035  csbnest1g 2037  csbco3g 2040  sbcco3g 2041

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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