Theorem mo2n 1928

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 1737	. 2 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3		alnex 1388	. . 3 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
4		nfs1v 1815	. . . . . 6 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
5	4	nfn 1548	. . . . 5 ⊢ Ⅎ𝑥 ¬ [𝑦 / 𝑥]𝜑
6	1	nfn 1548	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑
7		sbequ1 1651	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
8	7	equcoms 1594	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑))
9	8	con3d 561	. . . . 5 ⊢ (𝑦 = 𝑥 → (¬ [𝑦 / 𝑥]𝜑 → ¬ 𝜑))
10	5, 6, 9	cbv3 1630	. . . 4 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 → ∀𝑥 ¬ 𝜑)
11		pm2.21 547	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝑥 = 𝑦))
12	11	alimi 1344	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝑥 = 𝑦))
13		19.8a 1482	. . . 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 125	. 2 ⊢ (¬ ∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
16	2, 15	sylnbi 603	1 ⊢ (¬ ∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646

This theorem is referenced by:  modc  1943