Theorem csbeq2d 2955

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 2157	. . . 4 |- ( ph -> ( y e. B <-> y e. C ) )
4	1, 3	sbcbid 2896	. . 3 |- ( ph -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
5	4	abbidv 2205	. 2 |- ( ph -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb 2934	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb 2934	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5, 6, 7	3eqtr4g 2145	1 |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1289   F/wnf 1394   e. wcel 1438   {cab 2074   [.wsbc 2840   [_csb 2933

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-sbc 2841  df-csb 2934

This theorem is referenced by:  csbeq2dv  2956