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Theorem nmo 29655
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 2616 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32notbii 309 . 2 (¬ ∃*𝑥𝜑 ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
4 alnex 1855 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 exnal 1903 . . . 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 2979 . . . . . . 7 𝑥 = 𝑦𝑥𝑦)
98anbi2i 732 . . . . . 6 ((𝜑 ∧ ¬ 𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
106, 9bitri 264 . . . . 5 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
1110exbii 1923 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
125, 11bitr3i 266 . . 3 (¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
1312albii 1896 . 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 1630  wex 1853  wnf 1857  ∃*wmo 2608  wne 2932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612  df-ne 2933
This theorem is referenced by: (None)
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