Theorem sbceqbid 3442

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 2741	. . 3 |- ( ph -> { x | ps } = { x | ch } )
4	1, 3	eleq12d 2695	. 2 |- ( ph -> ( A e. { x | ps } <-> B e. { x | ch } ) )
5		df-sbc 3436	. 2 |- ( [. A / x ]. ps <-> A e. { x | ps } )
6		df-sbc 3436	. 2 |- ( [. B / x ]. ch <-> B e. { x | ch } )
7	4, 5, 6	3bitr4g 303	1 |- ( ph -> ( [. A / x ]. ps <-> [. B / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  fpwwe2cbv  9452  fpwwe2lem2  9454  fpwwe2lem3  9455  fi1uzind  13279  fi1uzindOLD  13285  isprs  16930  isdrs  16934  istos  17035  isdlat  17193  issrg  18507  islmod  18867  fdc  33541  hdmap1ffval  37085  hdmap1fval  37086  hdmapffval  37118  hdmapfval  37119  hgmapffval  37177  hgmapfval  37178  sbccomieg  37357  rexrabdioph  37358