Theorem raleqbii 2644

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	⊢ (ψ ↔ χ)

Assertion

Ref	Expression
raleqbii	⊢ (∀x ∈ A ψ ↔ ∀x ∈ B χ)

Proof of Theorem raleqbii

Step	Hyp	Ref	Expression
1		raleqbii.1	. . . 4 ⊢ A = B
2	1	eleq2i 2417	. . 3 ⊢ (x ∈ A ↔ x ∈ B)
3		raleqbii.2	. . 3 ⊢ (ψ ↔ χ)
4	2, 3	imbi12i 316	. 2 ⊢ ((x ∈ A → ψ) ↔ (x ∈ B → χ))
5	4	ralbii2 2642	1 ⊢ (∀x ∈ A ψ ↔ ∀x ∈ B χ)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614

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 2619

This theorem is referenced by: (None)