Theorem equviniOLD

Description: Obsolete version of equvini as of 16-Sep-2023. (Contributed by NM, 10-Jan-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 15-Sep-2018) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		equtr	⊢ ( 𝑧 = 𝑥 → ( 𝑥 = 𝑦 → 𝑧 = 𝑦 ) )
2		equeuclr	⊢ ( 𝑧 = 𝑦 → ( 𝑥 = 𝑦 → 𝑥 = 𝑧 ) )
3	2	anc2ri	⊢ ( 𝑧 = 𝑦 → ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) ) )
4	1 3	syli	⊢ ( 𝑧 = 𝑥 → ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) ) )
5		19.8a	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) → ∃ 𝑧 ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) )
6	4 5	syl6	⊢ ( 𝑧 = 𝑥 → ( 𝑥 = 𝑦 → ∃ 𝑧 ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) ) )
7		ax13	⊢ ( ¬ 𝑧 = 𝑥 → ( 𝑥 = 𝑦 → ∀ 𝑧 𝑥 = 𝑦 ) )
8		ax6e	⊢ ∃ 𝑧 𝑧 = 𝑦
9	8 3	eximii	⊢ ∃ 𝑧 ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) )
10	9	19.35i	⊢ ( ∀ 𝑧 𝑥 = 𝑦 → ∃ 𝑧 ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) )
11	7 10	syl6	⊢ ( ¬ 𝑧 = 𝑥 → ( 𝑥 = 𝑦 → ∃ 𝑧 ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) ) )
12	6 11	pm2.61i	⊢ ( 𝑥 = 𝑦 → ∃ 𝑧 ( 𝑥 = 𝑧 ∧ 𝑧 = 𝑦 ) )