Metamath Proof Explorer


Theorem dveeq1-o

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq1 using ax-c11 . (Contributed by NM, 2-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion dveeq1-o ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) )

Proof

Step Hyp Ref Expression
1 ax-5 ( 𝑤 = 𝑧 → ∀ 𝑥 𝑤 = 𝑧 )
2 ax-5 ( 𝑦 = 𝑧 → ∀ 𝑤 𝑦 = 𝑧 )
3 equequ1 ( 𝑤 = 𝑦 → ( 𝑤 = 𝑧𝑦 = 𝑧 ) )
4 1 2 3 dvelimf-o ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) )