Theorem sbceqbid 2967

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 2293	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	1, 3	eleq12d 2246	. 2 |- ( ph -> ( A e. { x | ps } <-> B e. { x | ch } ) )
5		df-sbc 2961	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 2961	. 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 2146   {cab 2161   [.wsbc 2960

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-sbc 2961

This theorem is referenced by:  issrg  12941