Theorem raleqbii 2645

Description: Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
raleqbii.1	|- A = B
raleqbii.2	|- (ps <-> ch)

Assertion

Ref	Expression
raleqbii	|- (A.x e. A ps <-> A.x e. B ch)

Proof of Theorem raleqbii

Step	Hyp	Ref	Expression
1		raleqbii.1	. . . 4 |- A = B
2	1	eleq2i 2417	. . 3 |- (x e. A <-> x e. B)
3		raleqbii.2	. . 3 |- (ps <-> ch)
4	2, 3	imbi12i 316	. 2 |- ((x e. A -> ps) <-> (x e. B -> ch))
5	4	ralbii2 2643	1 |- (A.x e. A ps <-> A.x e. B ch)

Syntax hints:   <-> wb 176   = wceq 1642   e. wcel 1710  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-ral 2620

This theorem is referenced by: (None)