Theorem dveel1

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	dveel1	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 ∈ 𝑧 → ∀ 𝑥 𝑦 ∈ 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		elequ1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) )
2	1	dvelimv	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 ∈ 𝑧 → ∀ 𝑥 𝑦 ∈ 𝑧 ) )