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Theorem equvin 1743
 Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
equvin (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvin
StepHypRef Expression
1 equvini 1641 . 2 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
2 ax-17 1419 . . 3 (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3 equtr 1595 . . . 4 (𝑥 = 𝑧 → (𝑧 = 𝑦𝑥 = 𝑦))
43imp 115 . . 3 ((𝑥 = 𝑧𝑧 = 𝑦) → 𝑥 = 𝑦)
52, 4exlimih 1484 . 2 (∃𝑧(𝑥 = 𝑧𝑧 = 𝑦) → 𝑥 = 𝑦)
61, 5impbii 117 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃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: (None)
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