Theorem csbie2g 3016

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2911 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 2972	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
2		csbie2g.1	. . . . 5 |- ( x = y -> B = C )
3	2	eleq2d 2184	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
4		csbie2g.2	. . . . 5 |- ( y = A -> C = D )
5	4	eleq2d 2184	. . . 4 |- ( y = A -> ( z e. C <-> z e. D ) )
6	3, 5	sbcie2g 2910	. . 3 |- ( A e. V -> ( [. A / x ]. z e. B <-> z e. D ) )
7	6	abbi1dv 2234	. 2 |- ( A e. V -> { z | [. A / x ]. z e. B } = D )
8	1, 7	syl5eq 2159	1 |- ( A e. V -> [_ A / x ]_ B = D )

Syntax hints:   -> wi 4   = wceq 1314   e. wcel 1463   {cab 2101   [.wsbc 2878   [_csb 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879  df-csb 2972

This theorem is referenced by: (None)