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Theorem 19.32 2088
Description: Theorem 19.32 of [Margaris] p. 90. See 19.32v 1856 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
Hypothesis
Ref Expression
19.32.1 𝑥𝜑
Assertion
Ref Expression
19.32 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))

Proof of Theorem 19.32
StepHypRef Expression
1 19.32.1 . . . 4 𝑥𝜑
21nfn 1768 . . 3 𝑥 ¬ 𝜑
3219.21 2062 . 2 (∀𝑥𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
4 df-or 384 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
54albii 1737 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥𝜑𝜓))
6 df-or 384 . 2 ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
73, 5, 63bitr4i 291 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wal 1473  wnf 1699
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
This theorem is referenced by:  19.31  2089  2eu3  2543  axi12  2588
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