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Theorem nmo 28709
 Description: Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
Hypothesis
Ref Expression
nmo.1 𝑦𝜑
Assertion
Ref Expression
nmo (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nmo
StepHypRef Expression
1 nmo.1 . . . 4 𝑦𝜑
21mo2 2467 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32notbii 309 . 2 (¬ ∃*𝑥𝜑 ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
4 alnex 1697 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 exnal 1744 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑦))
6 pm4.61 441 . . . . . 6 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑 ∧ ¬ 𝑥 = 𝑦))
7 biid 250 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
87necon3bbii 2829 . . . . . . 7 𝑥 = 𝑦𝑥𝑦)
98anbi2i 726 . . . . . 6 ((𝜑 ∧ ¬ 𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
106, 9bitri 263 . . . . 5 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
1110exbii 1764 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
125, 11bitr3i 265 . . 3 (¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
1312albii 1737 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
143, 4, 133bitr2i 287 1 (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699  ∃*wmo 2459   ≠ wne 2780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-ne 2782 This theorem is referenced by: (None)
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