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Theorem ralbi 2562
 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 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))

Proof of Theorem ralbi
StepHypRef Expression
1 nfra1 2464 . 2 𝑥𝑥𝐴 (𝜑𝜓)
2 rsp 2478 . . 3 (∀𝑥𝐴 (𝜑𝜓) → (𝑥𝐴 → (𝜑𝜓)))
32imp 123 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
41, 3ralbida 2429 1 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 1480  ∀wral 2414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419 This theorem is referenced by:  uniiunlem  3180  iineq2  3825  ralrnmpt  5555  f1mpt  5665  mpo2eqb  5873  ralrnmpo  5878  cau3lem  10879
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