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Theorem nmo 29453
 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 2507 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32notbii 309 . 2 (¬ ∃*𝑥𝜑 ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
4 alnex 1746 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 exnal 1794 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑦))
6 pm4.61 441 . . . . . 6 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑 ∧ ¬ 𝑥 = 𝑦))
7 biid 251 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
87necon3bbii 2870 . . . . . . 7 𝑥 = 𝑦𝑥𝑦)
98anbi2i 730 . . . . . 6 ((𝜑 ∧ ¬ 𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
106, 9bitri 264 . . . . 5 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
1110exbii 1814 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
125, 11bitr3i 266 . . 3 (¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
1312albii 1787 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
143, 4, 133bitr2i 288 1 (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521  ∃wex 1744  Ⅎwnf 1748  ∃*wmo 2499   ≠ wne 2823 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-10 2059  ax-11 2074  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503  df-ne 2824 This theorem is referenced by: (None)
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