Theorem csbeq2d 2930

Description: Formula-building deduction rule 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 2148	. . . 4 |- ( ph -> ( y e. B <-> y e. C ) )
4	1, 3	sbcbid 2871	. . 3 |- ( ph -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
5	4	abbidv 2196	. 2 |- ( ph -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb 2909	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb 2909	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5, 6, 7	3eqtr4g 2138	1 |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1284   F/wnf 1389   e. wcel 1433   {cab 2067   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-sbc 2816  df-csb 2909

This theorem is referenced by:  csbeq2dv  2931