Theorem nmo 30365

Description: Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)

Hypothesis

Ref	Expression
nmo.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
nmo	⊢ (¬ ∃*𝑥𝜑 ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nmo

Step	Hyp	Ref	Expression
1		nmo.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	mof 2581	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
3	2	notbii 323	. 2 ⊢ (¬ ∃*𝑥𝜑 ↔ ¬ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
4		alnex 1783	. 2 ⊢ (∀𝑦 ¬ ∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ¬ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
5		exnal 1828	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 → 𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑 → 𝑥 = 𝑦))
6		pm4.61 408	. . . . . 6 ⊢ (¬ (𝜑 → 𝑥 = 𝑦) ↔ (𝜑 ∧ ¬ 𝑥 = 𝑦))
7		biid 264	. . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
8	7	necon3bbii 2998	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
9	8	anbi2i 625	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 ≠ 𝑦))
10	6, 9	bitri 278	. . . . 5 ⊢ (¬ (𝜑 → 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 ≠ 𝑦))
11	10	exbii 1849	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 → 𝑥 = 𝑦) ↔ ∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦))
12	5, 11	bitr3i 280	. . 3 ⊢ (¬ ∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦))
13	12	albii 1821	. 2 ⊢ (∀𝑦 ¬ ∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦))
14	3, 4, 13	3bitr2i 302	1 ⊢ (¬ ∃*𝑥𝜑 ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦))

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)