Theorem ralbi 2439

Description: Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)

Assertion

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

Proof of Theorem ralbi

Step	Hyp	Ref	Expression
1		nfra1 2349	. 2 ⊢ Ⅎx∀x ∈ A (φ ↔ ψ)
2		rsp 2363	. . 3 ⊢ (∀x ∈ A (φ ↔ ψ) → (x ∈ A → (φ ↔ ψ)))
3	2	imp 115	. 2 ⊢ ((∀x ∈ A (φ ↔ ψ) ∧ x ∈ A) → (φ ↔ ψ))
4	1, 3	ralbida 2314	1 ⊢ (∀x ∈ A (φ ↔ ψ) → (∀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  ax-4 1397  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305

This theorem is referenced by:  uniiunlem  3022  iineq2  3665  ralrnmpt  5252  f1mpt  5353  mpt22eqb  5552  ralrnmpt2  5557