Theorem rabimbieq 34016

Description: Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.)

Hypotheses

Ref	Expression
rabimbieq.1	|- B = { x e. A | ph }
rabimbieq.2	|- ( x e. A -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem rabimbieq

Step	Hyp	Ref	Expression
1		rabimbieq.1	. 2 |- B = { x e. A | ph }
2		rabimbieq.2	. . 3 |- ( x e. A -> ( ph <-> ps ) )
3	2	rabbiia 3185	. 2 |- { x e. A | ph } = { x e. A | ps }
4	1, 3	eqtri 2644	1 |- B = { x e. A | ps }

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   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:  abeqinbi  34018