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Metamath Proof Explorer

Theorem dveel2

Description: Quantifier introduction when one pair of variables is disjoint. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jan-2002) (New usage is discouraged.)

		Ref	Expression
	Assertion	dveel2	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑦 → ∀ 𝑥 𝑧 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		elequ2	⊢ ( 𝑤 = 𝑦 → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦 ) )
2	1	dvelimv	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑦 → ∀ 𝑥 𝑧 ∈ 𝑦 ) )