Theorem rabeqbidva 2756

Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

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

Assertion

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

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hints:   ps( x)   ch( x)

Proof of Theorem rabeqbidva

Step	Hyp	Ref	Expression
1		rabeqbidva.2	. . 3 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
2	1	rabbidva 2748	. 2 |- ( ph -> { x e. A | ps } = { x e. A | ch } )
3		rabeqbidva.1	. . 3 |- ( ph -> A = B )
4		rabeq 2752	. . 3 |- ( A = B -> { x e. A | ch } = { x e. B | ch } )
5	3, 4	syl 14	. 2 |- ( ph -> { x e. A | ch } = { x e. B | ch } )
6	2, 5	eqtrd 2226	1 |- ( ph -> { x e. A | ps } = { x e. B | ch } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   {crab 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481

This theorem is referenced by: (None)