Theorem equvinvOLD 2130

Description: Obsolete version of equvinv 2129 as of 12-Jul-2022. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2185, ax-13 2375. (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.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinvOLD

Step	Hyp	Ref	Expression
1		ax6ev 2074	. . 3 ⊢ ∃𝑧 𝑧 = 𝑥
2		equtrr 2121	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
3	2	ancld 547	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥 ∧ 𝑧 = 𝑦)))
4	3	eximdv 2013	. . 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 2115	. . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → 𝑥 = 𝑦))
7	6	imp 396	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
8	7	exlimiv 2026	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦)
9	5, 8	impbii 201	1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))

Syntax hints:   ↔ wb 198   ∧ wa 385  ∃wex 1875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876

This theorem is referenced by: (None)