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Theorem ralnralall 4521
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 3109 . 2 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑))
2 pm3.24 402 . . . . 5 ¬ (𝜑 ∧ ¬ 𝜑)
32bifal 1553 . . . 4 ((𝜑 ∧ ¬ 𝜑) ↔ ⊥)
43ralbii 3091 . . 3 (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) ↔ ∀𝑥𝐴 ⊥)
5 r19.3rzv 4505 . . . 4 (𝐴 ≠ ∅ → (⊥ ↔ ∀𝑥𝐴 ⊥))
6 falim 1554 . . . 4 (⊥ → 𝜓)
75, 6biimtrrdi 254 . . 3 (𝐴 ≠ ∅ → (∀𝑥𝐴 ⊥ → 𝜓))
84, 7biimtrid 242 . 2 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑 ∧ ¬ 𝜑) → 𝜓))
91, 8biimtrrid 243 1 (𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 ¬ 𝜑) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wfal 1549  wne 2938  wral 3059  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-ne 2939  df-ral 3060  df-dif 3966  df-nul 4340
This theorem is referenced by: (None)
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