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Theorem r19.32 41691
Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3221. (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 1933 . . 3 𝑥 ¬ 𝜑
32r19.21 3094 . 2 (∀𝑥𝐴𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
4 df-or 384 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
54ralbii 3118 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴𝜑𝜓))
6 df-or 384 . 2 ((𝜑 ∨ ∀𝑥𝐴 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝐴 𝜓))
73, 5, 63bitr4i 292 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wnf 1857  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1854  df-nf 1859  df-ral 3055
This theorem is referenced by:  2reu3  41712
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