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Theorem 19.32 2099
Description: Theorem 19.32 of [Margaris] p. 90. See 19.32v 1866 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 1781 . . 3 𝑥 ¬ 𝜑
3219.21 2073 . 2 (∀𝑥𝜑𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
4 df-or 385 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
54albii 1744 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥𝜑𝜓))
6 df-or 385 . 2 ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))
73, 5, 63bitr4i 292 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wal 1478  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by:  19.31  2100  2eu3  2554  axi12  2599
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