Theorem csbeq2d 3084

Description: Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
csbeq2d.1	|- F/ x ph
csbeq2d.2	|- ( ph -> B = C )

Assertion

Ref	Expression
csbeq2d	|- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )

Proof of Theorem csbeq2d

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq2d.1	. . . 4 |- F/ x ph
2		csbeq2d.2	. . . . 5 |- ( ph -> B = C )
3	2	eleq2d 2247	. . . 4 |- ( ph -> ( y e. B <-> y e. C ) )
4	1, 3	sbcbid 3022	. . 3 |- ( ph -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
5	4	abbidv 2295	. 2 |- ( ph -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb 3060	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb 3060	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5, 6, 7	3eqtr4g 2235	1 |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1353   F/wnf 1460   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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  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-sbc 2965  df-csb 3060

This theorem is referenced by:  csbeq2dv  3085