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Theorem r19.32r 2578
 Description: One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
Hypothesis
Ref Expression
r19.32r.1 𝑥𝜑
Assertion
Ref Expression
r19.32r ((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))

Proof of Theorem r19.32r
StepHypRef Expression
1 r19.32r.1 . . . 4 𝑥𝜑
2 orc 701 . . . . 5 (𝜑 → (𝜑𝜓))
32a1d 22 . . . 4 (𝜑 → (𝑥𝐴 → (𝜑𝜓)))
41, 3alrimi 1502 . . 3 (𝜑 → ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
5 df-ral 2421 . . . 4 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
6 olc 700 . . . . . 6 (𝜓 → (𝜑𝜓))
76imim2i 12 . . . . 5 ((𝑥𝐴𝜓) → (𝑥𝐴 → (𝜑𝜓)))
87alimi 1431 . . . 4 (∀𝑥(𝑥𝐴𝜓) → ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
95, 8sylbi 120 . . 3 (∀𝑥𝐴 𝜓 → ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
104, 9jaoi 705 . 2 ((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
11 df-ral 2421 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
1210, 11sylibr 133 1 ((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 697  ∀wal 1329  Ⅎwnf 1436   ∈ wcel 1480  ∀wral 2416 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 698  ax-5 1423  ax-gen 1425  ax-4 1487 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2421 This theorem is referenced by:  r19.32vr  2579
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