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Theorem nmo 30051
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 2576 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32notbii 312 . 2 (¬ ∃*𝑥𝜑 ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
4 alnex 1745 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ¬ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
5 exnal 1790 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑𝑥 = 𝑦))
6 pm4.61 396 . . . . . 6 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑 ∧ ¬ 𝑥 = 𝑦))
7 biid 253 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
87necon3bbii 3007 . . . . . . 7 𝑥 = 𝑦𝑥𝑦)
98anbi2i 614 . . . . . 6 ((𝜑 ∧ ¬ 𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
106, 9bitri 267 . . . . 5 (¬ (𝜑𝑥 = 𝑦) ↔ (𝜑𝑥𝑦))
1110exbii 1811 . . . 4 (∃𝑥 ¬ (𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
125, 11bitr3i 269 . . 3 (¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∃𝑥(𝜑𝑥𝑦))
1312albii 1783 . 2 (∀𝑦 ¬ ∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
143, 4, 133bitr2i 291 1 (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wal 1506  wex 1743  wnf 1747  ∃*wmo 2546  wne 2960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-10 2080  ax-11 2094  ax-12 2107
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-ex 1744  df-nf 1748  df-mo 2548  df-ne 2961
This theorem is referenced by: (None)
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