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Theorem ralralimp 40309
 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 938 . . . 4 (𝜑 → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
21adantr 480 . . 3 ((𝜑𝐴 ≠ ∅) → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
32ralimdv 2946 . 2 ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → ∀𝑥𝐴 𝜏))
4 rspn0 3892 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜏𝜏))
54adantl 481 . 2 ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 𝜏𝜏))
63, 5syld 46 1 ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ≠ wne 2780  ∀wral 2896  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by: (None)
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