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Theorem equvini 1657
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 = 𝑧𝑧 = 𝑦))

Proof of Theorem equvini
StepHypRef Expression
1 ax12or 1419 . 2 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 equcomi 1608 . . . . . . 7 (𝑧 = 𝑥𝑥 = 𝑧)
32alimi 1360 . . . . . 6 (∀𝑧 𝑧 = 𝑥 → ∀𝑧 𝑥 = 𝑧)
4 a9e 1602 . . . . . 6 𝑧 𝑧 = 𝑦
53, 4jctir 300 . . . . 5 (∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑥 = 𝑧 ∧ ∃𝑧 𝑧 = 𝑦))
65a1d 22 . . . 4 (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → (∀𝑧 𝑥 = 𝑧 ∧ ∃𝑧 𝑧 = 𝑦)))
7 19.29 1527 . . . 4 ((∀𝑧 𝑥 = 𝑧 ∧ ∃𝑧 𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
86, 7syl6 33 . . 3 (∀𝑧 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
9 a9e 1602 . . . . . . . 8 𝑧 𝑧 = 𝑥
102eximi 1507 . . . . . . . 8 (∃𝑧 𝑧 = 𝑥 → ∃𝑧 𝑥 = 𝑧)
119, 10ax-mp 7 . . . . . . 7 𝑧 𝑥 = 𝑧
12112a1i 27 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∃𝑧 𝑥 = 𝑧))
1312anc2ri 317 . . . . 5 (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑧 𝑥 = 𝑧 ∧ ∀𝑧 𝑧 = 𝑦)))
14 19.29r 1528 . . . . 5 ((∃𝑧 𝑥 = 𝑧 ∧ ∀𝑧 𝑧 = 𝑦) → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
1513, 14syl6 33 . . . 4 (∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
16 ax-8 1411 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
1716anc2li 316 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦)))
1817equcoms 1610 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑥 = 𝑦 → (𝑥 = 𝑧𝑧 = 𝑦)))
1918com12 30 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑥 = 𝑧𝑧 = 𝑦)))
2019alimi 1360 . . . . . . . 8 (∀𝑧 𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥 → (𝑥 = 𝑧𝑧 = 𝑦)))
21 exim 1506 . . . . . . . 8 (∀𝑧(𝑧 = 𝑥 → (𝑥 = 𝑧𝑧 = 𝑦)) → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
2220, 21syl 14 . . . . . . 7 (∀𝑧 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
239, 22mpi 15 . . . . . 6 (∀𝑧 𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
2423imim2i 12 . . . . 5 ((𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
2524sps 1446 . . . 4 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
2615, 25jaoi 646 . . 3 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
278, 26jaoi 646 . 2 ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦)))
281, 27ax-mp 7 1 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 639  wal 1257   = wceq 1259  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  sbequi  1736  equvin  1759
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