Theorem equvin 1877

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 1772	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))
2		ax-17 1540	. . 3 ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3		equtr 1723	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑧 = 𝑦 → 𝑥 = 𝑦))
4	3	imp 124	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
5	2, 4	exlimih 1607	. 2 ⊢ (∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
6	1, 5	impbii 126	1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)