Theorem csbie2g 3564

Description: Conversion of implicit substitution to explicit class substitution. This version of csbie 3559 avoids a disjointness condition on x , A and x , D 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 3534	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
2		csbie2g.1	. . . . 5 |- ( x = y -> B = C )
3	2	eleq2d 2687	. . . 4 |- ( x = y -> ( z e. B <-> z e. C ) )
4		csbie2g.2	. . . . 5 |- ( y = A -> C = D )
5	4	eleq2d 2687	. . . 4 |- ( y = A -> ( z e. C <-> z e. D ) )
6	3, 5	sbcie2g 3469	. . 3 |- ( A e. V -> ( [. A / x ]. z e. B <-> z e. D ) )
7	6	abbi1dv 2743	. 2 |- ( A e. V -> { z | [. A / x ]. z e. B } = D )
8	1, 7	syl5eq 2668	1 |- ( A e. V -> [_ A / x ]_ B = D )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by: (None)