Theorem sbcbid 3086

Description: Formula-building deduction 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 2346	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	3	eleq2d 2299	. 2 |- ( ph -> ( A e. { x | ps } <-> A e. { x | ch } ) )
5		df-sbc 3029	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 3029	. 2 |- ( [. A / x ]. ch <-> A e. { x | ch } )
7	4, 5, 6	3bitr4g 223	1 |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1506   e. wcel 2200   {cab 2215   [.wsbc 3028

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029

This theorem is referenced by:  sbcbidv  3087  csbeq2d  3149  bezoutlemstep  12518