Theorem rabbia2 3187

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

Hypothesis

Ref	Expression
rabbia2.1	|- ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) )

Assertion

Ref	Expression
rabbia2	|- { x e. A | ps } = { x e. B | ch }

Proof of Theorem rabbia2

Step	Hyp	Ref	Expression
1		rabbia2.1	. . . 4 |- ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) )
2	1	a1i 11	. . 3 |- ( T. -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
3	2	rabbidva2 3186	. 2 |- ( T. -> { x e. A | ps } = { x e. B | ch } )
4	3	trud 1493	1 |- { x e. A | ps } = { x e. B | ch }

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   {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:  finsumvtxdg2ssteplem3  26443  smflim  40985  smflim2  41012  smflimsuplem1  41026  smflimsup  41034  sprvalpwn0  41733