Theorem ralbi 2501

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 2409	. 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓)
2		rsp 2423	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)))
3	2	imp 122	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
4	1, 3	ralbida 2374	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ↔ wb 103   ∈ wcel 1438  ∀wral 2359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364

This theorem is referenced by:  uniiunlem  3109  iineq2  3747  ralrnmpt  5441  f1mpt  5550  mpt22eqb  5754  ralrnmpt2  5759  cau3lem  10543