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Theorem ralralimp 46065
Description: Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)
Assertion
Ref Expression
ralralimp ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜏,𝑥
Allowed substitution hint:   𝜃(𝑥)

Proof of Theorem ralralimp
StepHypRef Expression
1 ornld 1060 . . . 4 (𝜑 → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
21adantr 481 . . 3 ((𝜑𝐴 ≠ ∅) → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
32ralimdv 3169 . 2 ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → ∀𝑥𝐴 𝜏))
4 rspn0 4352 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜏𝜏))
54adantl 482 . 2 ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 𝜏𝜏))
63, 5syld 47 1 ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845  wne 2940  wral 3061  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-ne 2941  df-ral 3062  df-dif 3951  df-nul 4323
This theorem is referenced by: (None)
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