Theorem bj-rabeqbida 32918

Description: Version of rabeqbidva 3196 with two dv conditions removed and the third replaced by a non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)

Hypotheses

Ref	Expression
bj-rabeqbida.nf	|- F/ x ph
bj-rabeqbida.1	|- ( ph -> A = B )
bj-rabeqbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem bj-rabeqbida

Step	Hyp	Ref	Expression
1		bj-rabeqbida.nf	. . 3 |- F/ x ph
2		bj-rabeqbida.2	. . 3 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	1, 2	bj-rabbida 32914	. 2 |- ( ph -> { x e. A | ps } = { x e. A | ch } )
4		bj-rabeqbida.1	. . 3 |- ( ph -> A = B )
5	1, 4	bj-rabeqd 32916	. 2 |- ( ph -> { x e. A | ch } = { x e. B | ch } )
6	3, 5	eqtrd 2656	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   {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: (None)