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Theorem ralbi 2462
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 2372 . 2 𝑥𝑥𝐴 (𝜑𝜓)
2 rsp 2386 . . 3 (∀𝑥𝐴 (𝜑𝜓) → (𝑥𝐴 → (𝜑𝜓)))
32imp 119 . 2 ((∀𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))
41, 3ralbida 2337 1 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by:  uniiunlem  3056  iineq2  3702  ralrnmpt  5337  f1mpt  5438  mpt22eqb  5638  ralrnmpt2  5643  cau3lem  9941
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