Theorem equvinv 1959

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2019, ax-13 2246. (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.)

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		ax6ev 1890	. . 3 ⊢ ∃𝑧 𝑧 = 𝑥
2		equtrr 1949	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
3	2	ancld 576	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥 ∧ 𝑧 = 𝑦)))
4	3	eximdv 1846	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦)))
5	1, 4	mpi 20	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))
6		ax7 1943	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → 𝑥 = 𝑦))
7	6	imp 445	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
8	7	exlimiv 1858	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
9	5, 8	impbii 199	1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  equvelv  1963  ax8  1996  ax9  2003  ax13  2249  wl-ax8clv1  33378  wl-ax8clv2  33381