Theorem sbceqbid 2970

Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)

Hypotheses

Ref	Expression
sbceqbid.1	|- ( ph -> A = B )
sbceqbid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)   B( x)

Proof of Theorem sbceqbid

Step	Hyp	Ref	Expression
1		sbceqbid.1	. . 3 |- ( ph -> A = B )
2		sbceqbid.2	. . . 4 |- ( ph -> ( ps <-> ch ) )
3	2	abbidv 2295	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	1, 3	eleq12d 2248	. 2 |- ( ph -> ( A e. { x | ps } <-> B e. { x | ch } ) )
5		df-sbc 2964	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 2964	. 2 |- ( [. B / x ]. ch <-> B e. { x | ch } )
7	4, 5, 6	3bitr4g 223	1 |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   {cab 2163   [.wsbc 2963

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2964

This theorem is referenced by:  issrg  13148  islmod  13381