Metamath Proof Explorer


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 → ( 𝑦𝑧 → ∀ 𝑥 𝑦𝑧 ) )

Proof

Step Hyp Ref Expression
1 elequ1 ( 𝑤 = 𝑦 → ( 𝑤𝑧𝑦𝑧 ) )
2 1 dvelimv ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦𝑧 → ∀ 𝑥 𝑦𝑧 ) )