Theorem csbie2g 3109

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2999 avoids a disjointness condition on x and A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Hypotheses

Ref	Expression
csbie2g.1	|- ( x = y -> B = C )
csbie2g.2	|- ( y = A -> C = D )

Assertion

Ref	Expression
csbie2g	|- ( A e. V -> [_ A / x ]_ B = D )

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

Allowed substitution hints:   A( x)   B( x)   C( y)   D( x)   V( x, y)

Proof of Theorem csbie2g

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3060	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
2		csbie2g.1	. . . . 5 |- ( x = y -> B = C )
3	2	eleq2d 2247	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
4		csbie2g.2	. . . . 5 |- ( y = A -> C = D )
5	4	eleq2d 2247	. . . 4 |- ( y = A -> ( z e. C <-> z e. D ) )
6	3, 5	sbcie2g 2998	. . 3 |- ( A e. V -> ( [. A / x ]. z e. B <-> z e. D ) )
7	6	abbi1dv 2297	. 2 |- ( A e. V -> { z | [. A / x ]. z e. B } = D )
8	1, 7	eqtrid 2222	1 |- ( A e. V -> [_ A / x ]_ B = D )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {cab 2163   [.wsbc 2964   [_csb 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-csb 3060

This theorem is referenced by: (None)