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Theorem nmo 30365
 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
21mof 2581 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32notbii 323 . 2 (¬ ∃*𝑥𝜑 ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
4 alnex 1783 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 exnal 1828 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑦))
6 pm4.61 408 . . . . . 6 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑 ∧ ¬ 𝑥 = 𝑦))
7 biid 264 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
87necon3bbii 2998 . . . . . . 7 𝑥 = 𝑦𝑥𝑦)
98anbi2i 625 . . . . . 6 ((𝜑 ∧ ¬ 𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
106, 9bitri 278 . . . . 5 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
1110exbii 1849 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
125, 11bitr3i 280 . . 3 (¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
1312albii 1821 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
143, 4, 133bitr2i 302 1 (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785  ∃*wmo 2555   ≠ wne 2951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-mo 2557  df-ne 2952 This theorem is referenced by: (None)
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