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Theorem ralralimp 43773
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 1057 . . . 4 (𝜑 → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
21adantr 484 . . 3 ((𝜑𝐴 ≠ ∅) → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
32ralimdv 3170 . 2 ((𝜑𝐴 ≠ ∅) → (∀𝑥𝐴 ((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → ∀𝑥𝐴 𝜏))
4 rspn0 4285 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 𝜏𝜏))
54adantl 485 . 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 399  wo 844  wne 3011  wral 3130  c0 4265
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-dif 3911  df-nul 4266
This theorem is referenced by: (None)
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