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Theorem r19.32v 3297
Description: Restricted quantifier version of 19.32v 1941. (Contributed by NM, 25-Nov-2003.)
Assertion
Ref Expression
r19.32v (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.32v
StepHypRef Expression
1 r19.21v 3145 . 2 (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
2 df-or 845 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32ralbii 3136 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜑𝜓))
4 df-or 845 . 2 ((𝜑 ∨ ∀𝑥𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
51, 3, 43bitr4i 306 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wo 844  wral 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-ral 3114
This theorem is referenced by:  iinun2  4961  iinuni  4986  axcontlem2  26763  axcontlem7  26768  disjnf  30337  lindslinindsimp2  44869
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