Theorem equvinvOLD 1949

Description: Obsolete version of equvinv 1946 as of 11-Apr-2021. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2006, ax-13 2234. (Revised by Wolf Lammen, 10-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinvOLD

Step	Hyp	Ref	Expression
1		equviniva 1947	. 2 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧))
2		equtrr 1936	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 → 𝑥 = 𝑦))
3	2	equcoms 1934	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → 𝑥 = 𝑦))
4	3	impcom 445	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
5	4	exlimiv 1845	. 2 ⊢ (∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)
6	1, 5	impbii 198	1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧))

Syntax hints:   → wi 4   ↔ 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: (None)