Theorem dveeq1

Description: Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jan-2002) Remove dependency on ax-11 . (Revised by Wolf Lammen, 8-Sep-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	dveeq1	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		nfeqf1	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑦 = 𝑧 )
2	1	nf5rd	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) )