Theorem sbceqbid 3058

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 2358	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	1, 3	eleq12d 2309	. 2 |- ( ph -> ( A e. { x | ps } <-> B e. { x | ch } ) )
5		df-sbc 3052	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 3052	. 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 1402   e. wcel 2209   {cab 2224   [.wsbc 3051

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052

This theorem is referenced by:  issrg  14243  islmod  14600