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Metamath Proof Explorer

Theorem dvelimv

Description: Similar to dvelim with first hypothesis replaced by a distinct variable condition. Usage of this theorem is discouraged because it depends on ax-13 . Check out dvelimhw for a version requiring fewer axioms. (Contributed by NM, 25-Jul-2015) (Proof shortened by Wolf Lammen, 30-Apr-2018) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	dvelimv.1	⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	Assertion	dvelimv	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝜓 → ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		dvelimv.1	⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		ax-5	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
3	2 1	dvelim	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝜓 → ∀ 𝑥 𝜓 ) )