Theorem rabbida 39274

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
rabbida.1	|- F/ x ph
rabbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
rabbida	|- ( ph -> { x e. A | ps } = { x e. A | ch } )

Proof of Theorem rabbida

Step	Hyp	Ref	Expression
1		rabbida.1	. . 3 |- F/ x ph
2		rabbida.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	ex 450	. . 3 |- ( ph -> ( x e. A -> ( ps <-> ch ) ) )
4	1, 3	ralrimi 2957	. 2 |- ( ph -> A. x e. A ( ps <-> ch ) )
5		rabbi 3120	. 2 |- ( A. x e. A ( ps <-> ch ) <-> { x e. A | ps } = { x e. A | ch } )
6	4, 5	sylib 208	1 |- ( ph -> { x e. A | ps } = { x e. A | ch } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   {crab 2916

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-ral 2917  df-rab 2921

This theorem is referenced by:  pimgtmnf  40932  smfpimltmpt  40955  smfpimltxrmpt  40967  smfpimgtmpt  40989  smfpimgtxrmpt  40992  smfrec  40996  smfsupmpt  41021  smfinflem  41023  smfinfmpt  41025