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Theorem ralnralall 40215
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 2950 . 2 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑))
2 pm3.24 921 . . . . 5 ¬ (𝜑 ∧ ¬ 𝜑)
32bifal 1487 . . . 4 ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
43ralbii 2867 . . 3 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥𝐴 ⊥)
5 r19.3rzv 3919 . . . 4 (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥𝐴 ⊥))
6 falim 1488 . . . 4 (⊥ → 𝜓)
75, 6syl6bir 242 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 ⊥ → 𝜓))
84, 7syl5bi 230 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓))
91, 8syl5bir 231 1 (𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wfal 1479  wne 2684  wral 2800  c0 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-v 3079  df-dif 3447  df-nul 3778
This theorem is referenced by: (None)
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