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Theorem ralnralall 4113
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 3093 . 2 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑))
2 pm3.24 944 . . . . 5 ¬ (𝜑 ∧ ¬ 𝜑)
32bifal 1537 . . . 4 ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
43ralbii 3009 . . 3 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥𝐴 ⊥)
5 r19.3rzv 4097 . . . 4 (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥𝐴 ⊥))
6 falim 1538 . . . 4 (⊥ → 𝜓)
75, 6syl6bir 244 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 ⊥ → 𝜓))
84, 7syl5bi 232 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓))
91, 8syl5bir 233 1 (𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wfal 1528  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-fal 1529  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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