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Metamath Proof Explorer

Theorem dveel2ALT

Description: Alternate proof of dveel2 using ax-c16 instead of ax-5 . (Contributed by NM, 10-May-2008) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	dveel2ALT	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑦 → ∀ 𝑥 𝑧 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		ax5el	⊢ ( 𝑧 ∈ 𝑤 → ∀ 𝑥 𝑧 ∈ 𝑤 )
2		ax5el	⊢ ( 𝑧 ∈ 𝑦 → ∀ 𝑤 𝑧 ∈ 𝑦 )
3		elequ2	⊢ ( 𝑤 = 𝑦 → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦 ) )
4	1 2 3	dvelimh	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑧 ∈ 𝑦 → ∀ 𝑥 𝑧 ∈ 𝑦 ) )