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Theorem r19.32 39816
 Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3064. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
Hypothesis
Ref Expression
r19.32.1 𝑥𝜑
Assertion
Ref Expression
r19.32 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))

Proof of Theorem r19.32
StepHypRef Expression
1 r19.32.1 . . . 4 𝑥𝜑
21nfn 1768 . . 3 𝑥 ¬ 𝜑
32r19.21 2939 . 2 (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
4 df-or 384 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
54ralbii 2963 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜑𝜓))
6 df-or 384 . 2 ((𝜑 ∨ ∀𝑥𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
73, 5, 63bitr4i 291 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382  Ⅎwnf 1699  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701  df-ral 2901 This theorem is referenced by:  2reu3  39837
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