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Theorem r19.32v 3231
 Description: Restricted quantifier version of 19.32v 2021. (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 3109 . 2 (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
2 df-or 837 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32ralbii 3129 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜑𝜓))
4 df-or 837 . 2 ((𝜑 ∨ ∀𝑥𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
51, 3, 43bitr4i 292 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 836  ∀wral 3061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991 This theorem depends on definitions:  df-bi 197  df-or 837  df-ex 1853  df-ral 3066 This theorem is referenced by:  iinun2  4720  iinuni  4743  axcontlem2  26066  axcontlem7  26071  disjnf  29722  lindslinindsimp2  42780
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