Theorem ralbiia 2332

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 169	. 2 ⊢ ((x ∈ A → φ) ↔ (x ∈ A → ψ))
3	2	ralbii2 2328	1 ⊢ (∀x ∈ A φ ↔ ∀x ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  ∀wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335

This theorem depends on definitions:  df-bi 110  df-ral 2305

This theorem is referenced by:  poinxp  4352  soinxp  4353  seinxp  4354  dffun8  4872  funcnv3  4904  fncnv  4908  fnres  4958  fvreseq  5214  isoini2  5401  smores  5848