Theorem equvinv 2041

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2145, ax-13 2373. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)

Assertion

Ref	Expression
equvinv	⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinv

Step	Hyp	Ref	Expression
1		equequ1 2037	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 ↔ 𝑥 = 𝑦))
2	1	equsexvw 2016	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦) ↔ 𝑥 = 𝑦)
3	2	bicomi 227	1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))

Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787

This theorem is referenced by:  ax8  2120  ax9  2128  ax13  2376  cossid  36254