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Theorem ralralimp 41619
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 958 . . . 4 (𝜑 → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
21adantr 480 . . 3 ((𝜑𝐴 ≠ ∅) → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
32ralimdv 2992 . 2 ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → ∀𝑥𝐴 𝜏))
4 rspn0 3967 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜏𝜏))
54adantl 481 . 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  wo 382  wa 383  wne 2823  wral 2941  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by: (None)
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