Theorem sbcbid 3090

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 2348	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	3	eleq2d 2301	. 2 |- ( ph -> ( A e. { x | ps } <-> A e. { x | ch } ) )
5		df-sbc 3033	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 3033	. 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 1509   e. wcel 2202   {cab 2217   [.wsbc 3032

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-sbc 3033

This theorem is referenced by:  sbcbidv  3091  csbeq2d  3153  bezoutlemstep  12648