Theorem ralbiia 2647

Description: Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)

Hypothesis

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

Assertion

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

Proof of Theorem ralbiia

Step	Hyp	Ref	Expression
1		ralbiia.1	. . 3 ⊢ (x ∈ A → (φ ↔ ψ))
2	1	pm5.74i 236	. 2 ⊢ ((x ∈ A → φ) ↔ (x ∈ A → ψ))
3	2	ralbii2 2643	1 ⊢ (∀x ∈ A φ ↔ ∀x ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∈ 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:  dffun8  5135  funcnv3  5158  fncnv  5159  fnres  5200  fvreseq  5399  isoini2  5499