Theorem nmo 32773

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 2597	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
3	2	notbii 323	. 2 ⊢ (¬ ∃*𝑥𝜑 ↔ ¬ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
4		alnex 1808	. 2 ⊢ (∀𝑦 ¬ ∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ¬ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
5		exnal 1854	. . . 4 ⊢ (∃𝑥 ¬ (𝜑 → 𝑥 = 𝑦) ↔ ¬ ∀𝑥(𝜑 → 𝑥 = 𝑦))
6		pm4.61 409	. . . . . 6 ⊢ (¬ (𝜑 → 𝑥 = 𝑦) ↔ (𝜑 ∧ ¬ 𝑥 = 𝑦))
7		biid 264	. . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
8	7	necon3bbii 3011	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 ↔ 𝑥 ≠ 𝑦)
9	8	anbi2i 634	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 ≠ 𝑦))
10	6, 9	bitri 278	. . . . 5 ⊢ (¬ (𝜑 → 𝑥 = 𝑦) ↔ (𝜑 ∧ 𝑥 ≠ 𝑦))
11	10	exbii 1875	. . . 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 1846	. 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 400  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810  ∃*wmo 2571   ≠ wne 2964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-mo 2573  df-ne 2965

This theorem is referenced by: (None)