Theorem ralbii2 2643

Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)

Hypothesis

Ref	Expression
ralbii2.1	⊢ ((x ∈ A → φ) ↔ (x ∈ B → ψ))

Assertion

Ref	Expression
ralbii2	⊢ (∀x ∈ A φ ↔ ∀x ∈ B ψ)

Proof of Theorem ralbii2

Step	Hyp	Ref	Expression
1		ralbii2.1	. . 3 ⊢ ((x ∈ A → φ) ↔ (x ∈ B → ψ))
2	1	albii 1566	. 2 ⊢ (∀x(x ∈ A → φ) ↔ ∀x(x ∈ B → ψ))
3		df-ral 2620	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
4		df-ral 2620	. 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ))
5	2, 3, 4	3bitr4i 268	1 ⊢ (∀x ∈ A φ ↔ ∀x ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  ∀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

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by:  raleqbii  2645  ralbiia  2647  ralrab  2999