Theorem sbcbid 2871

Description: Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbcbid.1	|- F/ x ph
sbcbid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
sbcbid	|- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Proof of Theorem sbcbid

Step	Hyp	Ref	Expression
1		sbcbid.1	. . . 4 |- F/ x ph
2		sbcbid.2	. . . 4 |- ( ph -> ( ps <-> ch ) )
3	1, 2	abbid 2195	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	3	eleq2d 2148	. 2 |- ( ph -> ( A e. { x | ps } <-> A e. { x | ch } ) )
5		df-sbc 2816	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 2816	. 2 |- ( [. A / x ]. ch <-> A e. { x | ch } )
7	4, 5, 6	3bitr4g 221	1 |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   e. wcel 1433   {cab 2067   [.wsbc 2815

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

This theorem is referenced by:  sbcbidv  2872  csbeq2d  2930  bezoutlemstep  10386