Theorem sbcbid 2961

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 2254	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	3	eleq2d 2207	. 2 |- ( ph -> ( A e. { x | ps } <-> A e. { x | ch } ) )
5		df-sbc 2905	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 2905	. 2 |- ( [. A / x ]. ch <-> A e. { x | ch } )
7	4, 5, 6	3bitr4g 222	1 |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1436   e. wcel 1480   {cab 2123   [.wsbc 2904

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-sbc 2905

This theorem is referenced by:  sbcbidv  2962  csbeq2d  3022  bezoutlemstep  11674