Theorem nexmo 2539

Description: Nonexistence implies uniqueness. (Contributed by BJ, 30-Sep-2022.) Avoid ax-11 2155. (Revised by Wolf Lammen, 16-Oct-2022.)

Assertion

Ref	Expression
nexmo	⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem nexmo

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2.21 123	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦))
2	1	alimi 1808	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦))
3	2	alrimiv 1925	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
4	3	19.2d 1975	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
5		alnex 1778	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6	5	bicomi 224	. 2 ⊢ (¬ ∃𝑥𝜑 ↔ ∀𝑥 ¬ 𝜑)
7		df-mo 2538	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
8	4, 6, 7	3imtr4i 292	1 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ∃wex 1776  ∃*wmo 2536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965

This theorem depends on definitions:  df-bi 207  df-ex 1777  df-mo 2538

This theorem is referenced by:  exmo  2540  moabs  2541  exmoeu  2579  moanimlem  2616  moexexlem  2624  mo2icl  3723  mosubopt  5520  dff3  7120  disjALTV0  38736