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Theorem r19.32r 2474
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 643 . . . . 5 (𝜑 → (𝜑𝜓))
32a1d 22 . . . 4 (𝜑 → (𝑥𝐴 → (𝜑𝜓)))
41, 3alrimi 1431 . . 3 (𝜑 → ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
5 df-ral 2328 . . . 4 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
6 olc 642 . . . . . 6 (𝜓 → (𝜑𝜓))
76imim2i 12 . . . . 5 ((𝑥𝐴𝜓) → (𝑥𝐴 → (𝜑𝜓)))
87alimi 1360 . . . 4 (∀𝑥(𝑥𝐴𝜓) → ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
95, 8sylbi 118 . . 3 (∀𝑥𝐴 𝜓 → ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
104, 9jaoi 646 . 2 ((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
11 df-ral 2328 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
1210, 11sylibr 141 1 ((𝜑 ∨ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639  wal 1257  wnf 1365  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-gen 1354  ax-4 1416
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328
This theorem is referenced by:  r19.32vr  2475
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