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Metamath Proof Explorer

Theorem dvelimnf

Description: Version of dvelim using "not free" notation. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 9-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvelimnf.1	⊢ Ⅎ 𝑥 𝜑
	dvelimnf.2	⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	dvelimnf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		dvelimnf.1	⊢ Ⅎ 𝑥 𝜑
2		dvelimnf.2	⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		nfv	⊢ Ⅎ 𝑧 𝜓
4	1 3 2	dvelimf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜓 )