Theorem equvinv 1946

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2006, ax-13 2234. (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 1877	. . 3 ⊢ ∃𝑧 𝑧 = 𝑥
2		equtrr 1936	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
3	2	ancld 574	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥 ∧ 𝑧 = 𝑦)))
4	3	eximdv 1833	. . 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 1930	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → 𝑥 = 𝑦))
7	6	imp 444	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
8	7	exlimiv 1845	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
9	5, 8	impbii 198	1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))

Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  equvelv  1950  ax8  1983  ax9  1990  ax13  2237