Theorem mo2n 2107

Description: There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)

Hypothesis

Ref	Expression
mon.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
mo2n	⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo2n

Step	Hyp	Ref	Expression
1		mon.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	sb8e 1905	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3		alnex 1547	. . 3 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
4		nfs1v 1992	. . . . . 6 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
5	4	nfn 1706	. . . . 5 ⊢ Ⅎ𝑥 ¬ [𝑦 / 𝑥]𝜑
6	1	nfn 1706	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑
7		sbequ1 1816	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
8	7	equcoms 1756	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑))
9	8	con3d 636	. . . . 5 ⊢ (𝑦 = 𝑥 → (¬ [𝑦 / 𝑥]𝜑 → ¬ 𝜑))
10	5, 6, 9	cbv3 1790	. . . 4 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 → ∀𝑥 ¬ 𝜑)
11		pm2.21 622	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦))
12	11	alimi 1503	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦))
13		19.8a 1638	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
14	10, 12, 13	3syl 17	. . 3 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
15	3, 14	sylbir 135	. 2 ⊢ (¬ ∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
16	2, 15	sylnbi 684	1 ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1395  Ⅎwnf 1508  ∃wex 1540  [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811

This theorem is referenced by:  modc  2123