Theorem equvin 1784

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 = 𝑧 ∧ 𝑧 = 𝑦))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvin

Step	Hyp	Ref	Expression
1		equvini 1681	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
2		ax-17 1459	. . 3 ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3		equtr 1635	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑧 = 𝑦 → 𝑥 = 𝑦))
4	3	imp 122	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
5	2, 4	exlimih 1524	. 2 ⊢ (∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
6	1, 5	impbii 124	1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))

Syntax hints:   ∧ wa 102   ↔ wb 103  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)