Theorem csbie2g 3099

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2989 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 3050	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
2		csbie2g.1	. . . . 5 |- ( x = y -> B = C )
3	2	eleq2d 2240	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
4		csbie2g.2	. . . . 5 |- ( y = A -> C = D )
5	4	eleq2d 2240	. . . 4 |- ( y = A -> ( z e. C <-> z e. D ) )
6	3, 5	sbcie2g 2988	. . 3 |- ( A e. V -> ( [. A / x ]. z e. B <-> z e. D ) )
7	6	abbi1dv 2290	. 2 |- ( A e. V -> { z | [. A / x ]. z e. B } = D )
8	1, 7	eqtrid 2215	1 |- ( A e. V -> [_ A / x ]_ B = D )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   {cab 2156   [.wsbc 2955   [_csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by: (None)