Theorem mo23 1923

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 1402	. . . . 5 ⊢ Ⅎy x = z
3	1, 2	nfim 1446	. . . 4 ⊢ Ⅎy(φ → x = z)
4	3	nfal 1450	. . 3 ⊢ Ⅎy∀x(φ → x = z)
5		nfv 1402	. . 3 ⊢ Ⅎz∀x(φ → x = y)
6		equequ2 1581	. . . . 5 ⊢ (z = y → (x = z ↔ x = y))
7	6	imbi2d 219	. . . 4 ⊢ (z = y → ((φ → x = z) ↔ (φ → x = y)))
8	7	albidv 1687	. . 3 ⊢ (z = y → (∀x(φ → x = z) ↔ ∀x(φ → x = y)))
9	4, 5, 8	cbvex 1621	. 2 ⊢ (∃z∀x(φ → x = z) ↔ ∃y∀x(φ → x = y))
10		nfs1v 1797	. . . . . . . 8 ⊢ Ⅎx[y / x]φ
11		nfv 1402	. . . . . . . 8 ⊢ Ⅎx y = z
12	10, 11	nfim 1446	. . . . . . 7 ⊢ Ⅎx([y / x]φ → y = z)
13		sbequ2 1634	. . . . . . . 8 ⊢ (x = y → ([y / x]φ → φ))
14		ax-8 1376	. . . . . . . 8 ⊢ (x = y → (x = z → y = z))
15	13, 14	imim12d 68	. . . . . . 7 ⊢ (x = y → ((φ → x = z) → ([y / x]φ → y = z)))
16	3, 12, 15	cbv3 1612	. . . . . 6 ⊢ (∀x(φ → x = z) → ∀y([y / x]φ → y = z))
17	16	ancli 306	. . . . 5 ⊢ (∀x(φ → x = z) → (∀x(φ → x = z) ∧ ∀y([y / x]φ → y = z)))
18	3	nfri 1393	. . . . . 6 ⊢ ((φ → x = z) → ∀y(φ → x = z))
19	12	nfri 1393	. . . . . 6 ⊢ (([y / x]φ → y = z) → ∀x([y / x]φ → y = z))
20	18, 19	aaanh 1460	. . . . 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 137	. . . 4 ⊢ (∀x(φ → x = z) → ∀x∀y((φ → x = z) ∧ ([y / x]φ → y = z)))
22		prth 326	. . . . . 6 ⊢ (((φ → x = z) ∧ ([y / x]φ → y = z)) → ((φ ∧ [y / x]φ) → (x = z ∧ y = z)))
23		equtr2 1579	. . . . . 6 ⊢ ((x = z ∧ y = z) → x = y)
24	22, 23	syl6 29	. . . . 5 ⊢ (((φ → x = z) ∧ ([y / x]φ → y = z)) → ((φ ∧ [y / x]φ) → x = y))
25	24	2alimi 1325	. . . 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 1471	. 2 ⊢ (∃z∀x(φ → x = z) → ∀x∀y((φ ∧ [y / x]φ) → x = y))
28	9, 27	sylbir 125	1 ⊢ (∃y∀x(φ → x = y) → ∀x∀y((φ ∧ [y / x]φ) → x = y))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1226  Ⅎwnf 1329  ∃wex 1362  [wsb 1627

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-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628

This theorem is referenced by:  modc  1925  eu2  1926  eu3h  1927