Theorem equveli 1773

Description: A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1772.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)

Assertion

Ref	Expression
equveli	⊢ (∀𝑧(𝑧 = 𝑥 ↔ 𝑧 = 𝑦) → 𝑥 = 𝑦)

Proof of Theorem equveli

Step	Hyp	Ref	Expression
1		albiim 1501	. 2 ⊢ (∀𝑧(𝑧 = 𝑥 ↔ 𝑧 = 𝑦) ↔ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥)))
2		ax12or 1522	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
3		equequ1 1726	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑥 ↔ 𝑥 = 𝑥))
4		equequ1 1726	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 ↔ 𝑥 = 𝑦))
5	3, 4	imbi12d 234	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑧 = 𝑥 → 𝑧 = 𝑦) ↔ (𝑥 = 𝑥 → 𝑥 = 𝑦)))
6	5	sps 1551	. . . . . . 7 ⊢ (∀𝑧 𝑧 = 𝑥 → ((𝑧 = 𝑥 → 𝑧 = 𝑦) ↔ (𝑥 = 𝑥 → 𝑥 = 𝑦)))
7	6	dral2 1745	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ↔ ∀𝑧(𝑥 = 𝑥 → 𝑥 = 𝑦)))
8		equid 1715	. . . . . . . . 9 ⊢ 𝑥 = 𝑥
9	8	a1bi 243	. . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ (𝑥 = 𝑥 → 𝑥 = 𝑦))
10	9	biimpri 133	. . . . . . 7 ⊢ ((𝑥 = 𝑥 → 𝑥 = 𝑦) → 𝑥 = 𝑦)
11	10	sps 1551	. . . . . 6 ⊢ (∀𝑧(𝑥 = 𝑥 → 𝑥 = 𝑦) → 𝑥 = 𝑦)
12	7, 11	biimtrdi 163	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → 𝑥 = 𝑦))
13	12	adantrd 279	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → ((∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥)) → 𝑥 = 𝑦))
14		equequ1 1726	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑦 ↔ 𝑦 = 𝑦))
15		equequ1 1726	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑦 = 𝑥))
16	14, 15	imbi12d 234	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝑧 = 𝑦 → 𝑧 = 𝑥) ↔ (𝑦 = 𝑦 → 𝑦 = 𝑥)))
17	16	sps 1551	. . . . . . . 8 ⊢ (∀𝑧 𝑧 = 𝑦 → ((𝑧 = 𝑦 → 𝑧 = 𝑥) ↔ (𝑦 = 𝑦 → 𝑦 = 𝑥)))
18	17	dral1 1744	. . . . . . 7 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥) ↔ ∀𝑦(𝑦 = 𝑦 → 𝑦 = 𝑥)))
19		equid 1715	. . . . . . . . 9 ⊢ 𝑦 = 𝑦
20		ax-4 1524	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑦 → 𝑦 = 𝑥) → (𝑦 = 𝑦 → 𝑦 = 𝑥))
21	19, 20	mpi 15	. . . . . . . 8 ⊢ (∀𝑦(𝑦 = 𝑦 → 𝑦 = 𝑥) → 𝑦 = 𝑥)
22		equcomi 1718	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
23	21, 22	syl 14	. . . . . . 7 ⊢ (∀𝑦(𝑦 = 𝑦 → 𝑦 = 𝑥) → 𝑥 = 𝑦)
24	18, 23	biimtrdi 163	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥) → 𝑥 = 𝑦))
25	24	adantld 278	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → ((∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥)) → 𝑥 = 𝑦))
26		hba1 1554	. . . . . . . . . 10 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ∀𝑧∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
27		hbequid 1527	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑥 → ∀𝑧 𝑥 = 𝑥)
28	27	a1i 9	. . . . . . . . . 10 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑥 → ∀𝑧 𝑥 = 𝑥))
29		ax-4 1524	. . . . . . . . . 10 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
30	26, 28, 29	hbimd 1587	. . . . . . . . 9 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ((𝑥 = 𝑥 → 𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥 → 𝑥 = 𝑦)))
31	30	a5i 1557	. . . . . . . 8 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ∀𝑧((𝑥 = 𝑥 → 𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥 → 𝑥 = 𝑦)))
32		equtr 1723	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑥 → 𝑧 = 𝑥))
33		ax-8 1518	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → 𝑥 = 𝑦))
34	32, 33	imim12d 74	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))
35	34	ax-gen 1463	. . . . . . . 8 ⊢ ∀𝑧(𝑧 = 𝑥 → ((𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))
36		19.26 1495	. . . . . . . . 9 ⊢ (∀𝑧(((𝑥 = 𝑥 → 𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥 → 𝑥 = 𝑦)) ∧ (𝑧 = 𝑥 → ((𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))) ↔ (∀𝑧((𝑥 = 𝑥 → 𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥 → 𝑥 = 𝑦)) ∧ ∀𝑧(𝑧 = 𝑥 → ((𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))))
37		spimth 1749	. . . . . . . . 9 ⊢ (∀𝑧(((𝑥 = 𝑥 → 𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥 → 𝑥 = 𝑦)) ∧ (𝑧 = 𝑥 → ((𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))) → (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))
38	36, 37	sylbir 135	. . . . . . . 8 ⊢ ((∀𝑧((𝑥 = 𝑥 → 𝑥 = 𝑦) → ∀𝑧(𝑥 = 𝑥 → 𝑥 = 𝑦)) ∧ ∀𝑧(𝑧 = 𝑥 → ((𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))) → (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))
39	31, 35, 38	sylancl 413	. . . . . . 7 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → (𝑥 = 𝑥 → 𝑥 = 𝑦)))
40	8, 39	mpii 44	. . . . . 6 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → 𝑥 = 𝑦))
41	40	adantrd 279	. . . . 5 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → ((∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥)) → 𝑥 = 𝑦))
42	25, 41	jaoi 717	. . . 4 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → ((∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥)) → 𝑥 = 𝑦))
43	13, 42	jaoi 717	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → ((∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥)) → 𝑥 = 𝑦))
44	2, 43	ax-mp 5	. 2 ⊢ ((∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ∧ ∀𝑧(𝑧 = 𝑦 → 𝑧 = 𝑥)) → 𝑥 = 𝑦)
45	1, 44	sylbi 121	1 ⊢ (∀𝑧(𝑧 = 𝑥 ↔ 𝑧 = 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  ∀wal 1362   = wceq 1364

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)