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Theorem 19.32r 1643
Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1642. (Contributed by Jim Kingdon, 28-Jul-2018.)
Hypothesis
Ref Expression
19.32r.1 𝑥𝜑
Assertion
Ref Expression
19.32r ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Proof of Theorem 19.32r
StepHypRef Expression
1 19.32r.1 . . 3 𝑥𝜑
2 orc 686 . . 3 (𝜑 → (𝜑𝜓))
31, 2alrimi 1487 . 2 (𝜑 → ∀𝑥(𝜑𝜓))
4 olc 685 . . 3 (𝜓 → (𝜑𝜓))
54alimi 1416 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
63, 5jaoi 690 1 ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 682  wal 1314  wnf 1421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-gen 1410  ax-4 1472
This theorem depends on definitions:  df-bi 116  df-nf 1422
This theorem is referenced by:  19.31r  1644
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