Theorem sbcbid 3008

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 2283	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	3	eleq2d 2236	. 2 |- ( ph -> ( A e. { x | ps } <-> A e. { x | ch } ) )
5		df-sbc 2952	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 2952	. 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 1448   e. wcel 2136   {cab 2151   [.wsbc 2951

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952

This theorem is referenced by:  sbcbidv  3009  csbeq2d  3070  bezoutlemstep  11930