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Theorem 19.32r 1613
Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1612. (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 666 . . 3 (𝜑 → (𝜑𝜓))
31, 2alrimi 1458 . 2 (𝜑 → ∀𝑥(𝜑𝜓))
4 olc 665 . . 3 (𝜓 → (𝜑𝜓))
54alimi 1387 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑𝜓))
63, 5jaoi 669 1 ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662  wal 1285  wnf 1392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-gen 1381  ax-4 1443
This theorem depends on definitions:  df-bi 115  df-nf 1393
This theorem is referenced by:  19.31r  1614
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