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Metamath Proof Explorer

Theorem dveeq2-o

Description: Quantifier introduction when one pair of variables is distinct. Version of dveeq2 using ax-c15 . (Contributed by NM, 2-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	dveeq2-o	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑧 = 𝑦 → ∀ 𝑥 𝑧 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		ax-5	⊢ ( 𝑧 = 𝑤 → ∀ 𝑥 𝑧 = 𝑤 )
2		ax-5	⊢ ( 𝑧 = 𝑦 → ∀ 𝑤 𝑧 = 𝑦 )
3		equequ2	⊢ ( 𝑤 = 𝑦 → ( 𝑧 = 𝑤 ↔ 𝑧 = 𝑦 ) )
4	1 2 3	dvelimf-o	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑧 = 𝑦 → ∀ 𝑥 𝑧 = 𝑦 ) )