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Theorem ralnralall 4270
Description: A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
Assertion
Ref Expression
ralnralall (𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑) → 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ralnralall
StepHypRef Expression
1 r19.26 3251 . 2 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑))
2 pm3.24 391 . . . . 5 ¬ (𝜑 ∧ ¬ 𝜑)
32bifal 1654 . . . 4 ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
43ralbii 3167 . . 3 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥𝐴 ⊥)
5 r19.3rzv 4256 . . . 4 (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥𝐴 ⊥))
6 falim 1655 . . . 4 (⊥ → 𝜓)
75, 6syl6bir 245 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 ⊥ → 𝜓))
84, 7syl5bi 233 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓))
91, 8syl5bir 234 1 (𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wfal 1650  wne 2977  wral 3095  c0 4113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-v 3392  df-dif 3769  df-nul 4114
This theorem is referenced by: (None)
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