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Theorem equvin 1843
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 1738 . 2 (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
2 ax-17 1506 . . 3 (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3 equtr 1689 . . . 4 (𝑥 = 𝑧 → (𝑧 = 𝑦𝑥 = 𝑦))
43imp 123 . . 3 ((𝑥 = 𝑧𝑧 = 𝑦) → 𝑥 = 𝑦)
52, 4exlimih 1573 . 2 (∃𝑧(𝑥 = 𝑧𝑧 = 𝑦) → 𝑥 = 𝑦)
61, 5impbii 125 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-i12 1487  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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