Theorem equvini 1641

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦 (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
equvini	⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))

Proof of Theorem equvini

Step	Hyp	Ref	Expression
1		ax12or 1403	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		equcomi 1592	. . . . . . 7 ⊢ (𝑧 = 𝑥 → 𝑥 = 𝑧)
3	2	alimi 1344	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → ∀𝑧 𝑥 = 𝑧)
4		a9e 1586	. . . . . 6 ⊢ ∃𝑧 𝑧 = 𝑦
5	3, 4	jctir 296	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑥 = 𝑧 ∧ ∃𝑧 𝑧 = 𝑦))
6	5	a1d 22	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → (∀𝑧 𝑥 = 𝑧 ∧ ∃𝑧 𝑧 = 𝑦)))
7		19.29 1511	. . . 4 ⊢ ((∀𝑧 𝑥 = 𝑧 ∧ ∃𝑧 𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
8	6, 7	syl6 29	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
9		a9e 1586	. . . . . . . 8 ⊢ ∃𝑧 𝑧 = 𝑥
10	2	eximi 1491	. . . . . . . 8 ⊢ (∃𝑧 𝑧 = 𝑥 → ∃𝑧 𝑥 = 𝑧)
11	9, 10	ax-mp 7	. . . . . . 7 ⊢ ∃𝑧 𝑥 = 𝑧
12	11	2a1i 24	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∃𝑧 𝑥 = 𝑧))
13	12	anc2ri 313	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑧 𝑥 = 𝑧 ∧ ∀𝑧 𝑧 = 𝑦)))
14		19.29r 1512	. . . . 5 ⊢ ((∃𝑧 𝑥 = 𝑧 ∧ ∀𝑧 𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
15	13, 14	syl6 29	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
16		ax-8 1395	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
17	16	anc2li 312	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
18	17	equcoms 1594	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
19	18	com12 27	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
20	19	alimi 1344	. . . . . . . 8 ⊢ (∀𝑧 𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
21		exim 1490	. . . . . . . 8 ⊢ (∀𝑧(𝑧 = 𝑥 → (𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
22	20, 21	syl 14	. . . . . . 7 ⊢ (∀𝑧 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
23	9, 22	mpi 15	. . . . . 6 ⊢ (∀𝑧 𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
24	23	imim2i 12	. . . . 5 ⊢ ((𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
25	24	sps 1430	. . . 4 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
26	15, 25	jaoi 636	. . 3 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
27	8, 26	jaoi 636	. 2 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)))
28	1, 27	ax-mp 7	1 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629  ∀wal 1241   = wceq 1243  ∃wex 1381

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-io 630  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  sbequi  1720  equvin  1743