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Metamath Proof Explorer

Theorem dveeq2

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) (Revised by NM, 20-Jul-2015) Remove dependency on ax-11 . (Revised by Wolf Lammen, 8-Sep-2018) (New usage is discouraged.)

		Ref	Expression
	Assertion	dveeq2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑧 = 𝑦 → ∀ 𝑥 𝑧 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		nfeqf2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝑧 = 𝑦 )
2	1	nf5rd	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑧 = 𝑦 → ∀ 𝑥 𝑧 = 𝑦 ) )