Theorem csbiegft 2000

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2002.)

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 1930	. . . 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 1788	. . . . . 6 |- z e. V
4		eleq1 1510	. . . . . . 7 |- (y = z -> (y e. C <-> z e. C))
5	4	albidv 1260	. . . . . . 7 |- (y = z -> (A.x y e. C <-> A.x z e. C))
6	4, 5	imbi12d 624	. . . . . 6 |- (y = z -> ((y e. C -> A.x y e. C) <-> (z e. C -> A.x z e. C)))
7	3, 6	cla4v 1841	. . . . 5 |- (A.y(y e. C -> A.x y e. C) -> (z e. C -> A.x z e. C))
8	7	19.20i 968	. . . 4 |- (A.xA.y(y e. C -> A.x y e. C) -> A.x(z e. C -> A.x z e. C))
9		eleq2 1511	. . . . . 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 968	. . . 4 |- (A.x(x = A -> B = C) -> A.x(x = A -> (z e. B <-> z e. C)))
12	1, 2, 8, 11	syl3an 865	. . 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 1555	. 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 1973	. 2 |- [_A / x]_B = {z | [A / x]z e. B}
15	13, 14	syl5eq 1495	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 772  A.wal 950   = wceq 1099   e. wcel 1105  [wsbc 1153  {cab 1440  [_csb 1972

This theorem is referenced by:  csbiegf 2002  csbnestglem 2006  csbnest1g 2008  csbco3g 2011  sbcco3g 2012

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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