Theorem mo23 2086

Description: An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)

Hypothesis

Ref	Expression
mo23.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
mo23	⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo23

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo23.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2		nfv 1542	. . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧
3	1, 2	nfim 1586	. . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧)
4	3	nfal 1590	. . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑧)
5		nfv 1542	. . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 → 𝑥 = 𝑦)
6		equequ2 1727	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
7	6	imbi2d 230	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜑 → 𝑥 = 𝑦)))
8	7	albidv 1838	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)))
9	4, 5, 8	cbvex 1770	. 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
10		nfs1v 1958	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
11		nfv 1542	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑦 = 𝑧
12	10, 11	nfim 1586	. . . . . . 7 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)
13		sbequ2 1783	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
14		ax-8 1518	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧))
15	13, 14	imim12d 74	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) → ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)))
16	3, 12, 15	cbv3 1756	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧))
17	16	ancli 323	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → (∀𝑥(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)))
18	3	nfri 1533	. . . . . 6 ⊢ ((𝜑 → 𝑥 = 𝑧) → ∀𝑦(𝜑 → 𝑥 = 𝑧))
19	12	nfri 1533	. . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧) → ∀𝑥([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧))
20	18, 19	aaanh 1600	. . . . 5 ⊢ (∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)) ↔ (∀𝑥(𝜑 → 𝑥 = 𝑧) ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)))
21	17, 20	sylibr 134	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)))
22		anim12 344	. . . . . 6 ⊢ (((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
23		equtr2 1725	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
24	22, 23	syl6 33	. . . . 5 ⊢ (((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
25	24	2alimi 1470	. . . 4 ⊢ (∀𝑥∀𝑦((𝜑 → 𝑥 = 𝑧) ∧ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑧)) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
26	21, 25	syl 14	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
27	26	exlimiv 1612	. 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
28	9, 27	sylbir 135	1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  Ⅎwnf 1474  ∃wex 1506  [wsb 1776

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  modc  2088  eu2  2089  eu3h  2090