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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 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))

Proof of Theorem ralbi
StepHypRef Expression
1 nfra1 2409 . 2 𝑥𝑥𝐴 (𝜑𝜓)
2 rsp 2423 . . 3 (∀𝑥𝐴 (𝜑𝜓) → (𝑥𝐴 → (𝜑𝜓)))
32imp 122 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
41, 3ralbida 2374 1 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))
Colors of variables: wff set class
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
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