Theorem rabbida2 39317

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
rabbida2.1	|- F/ x ph
rabbida2.2	|- ( ph -> A = B )
rabbida2.3	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
rabbida2	|- ( ph -> { x e. A | ps } = { x e. B | ch } )

Proof of Theorem rabbida2

Step	Hyp	Ref	Expression
1		rabbida2.1	. . 3 |- F/ x ph
2		rabbida2.2	. . . . 5 |- ( ph -> A = B )
3	2	eleq2d 2687	. . . 4 |- ( ph -> ( x e. A <-> x e. B ) )
4		rabbida2.3	. . . 4 |- ( ph -> ( ps <-> ch ) )
5	3, 4	anbi12d 747	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
6	1, 5	abbid 2740	. 2 |- ( ph -> { x | ( x e. A /\ ps ) } = { x | ( x e. B /\ ch ) } )
7		df-rab 2921	. 2 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
8		df-rab 2921	. 2 |- { x e. B | ch } = { x | ( x e. B /\ ch ) }
9	6, 7, 8	3eqtr4g 2681	1 |- ( ph -> { x e. A | ps } = { x e. B | ch } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   {cab 2608   {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-rab 2921

This theorem is referenced by:  smflimmpt  41016