Theorem equvinva

Description: A modified version of the forward implication of equvinv adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018)

	Ref	Expression
Assertion	equvinva	⊢ ( 𝑥 = 𝑦 → ∃ 𝑧 ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		ax6evr	⊢ ∃ 𝑧 𝑦 = 𝑧
2		equtr	⊢ ( 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → 𝑥 = 𝑧 ) )
3	2	ancrd	⊢ ( 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) ) )
4	3	eximdv	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑧 𝑦 = 𝑧 → ∃ 𝑧 ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) ) )
5	1 4	mpi	⊢ ( 𝑥 = 𝑦 → ∃ 𝑧 ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) )