Theorem csbeq2d 3032

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 2210	. . . 4 |- ( ph -> ( y e. B <-> y e. C ) )
4	1, 3	sbcbid 2970	. . 3 |- ( ph -> ( [. A / x ]. y e. B <-> [. A / x ]. y e. C ) )
5	4	abbidv 2258	. 2 |- ( ph -> { y | [. A / x ]. y e. B } = { y | [. A / x ]. y e. C } )
6		df-csb 3008	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
7		df-csb 3008	. 2 |- [_ A / x ]_ C = { y | [. A / x ]. y e. C }
8	5, 6, 7	3eqtr4g 2198	1 |- ( ph -> [_ A / x ]_ B = [_ A / x ]_ C )

Syntax hints:   -> wi 4   = wceq 1332   F/wnf 1437   e. wcel 1481   {cab 2126   [.wsbc 2913   [_csb 3007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-sbc 2914  df-csb 3008

This theorem is referenced by:  csbeq2dv  3033