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Metamath Proof Explorer

Theorem dvelimh

Description: Version of dvelim without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 . Check out dvelimhw for a version requiring fewer axioms. (Contributed by NM, 1-Oct-2002) (Proof shortened by Wolf Lammen, 11-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvelimh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	dvelimh.2	⊢ ( 𝜓 → ∀ 𝑧 𝜓 )
	dvelimh.3	⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	dvelimh	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝜓 → ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		dvelimh.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		dvelimh.2	⊢ ( 𝜓 → ∀ 𝑧 𝜓 )
3		dvelimh.3	⊢ ( 𝑧 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1	nf5i	⊢ Ⅎ 𝑥 𝜑
5	2	nf5i	⊢ Ⅎ 𝑧 𝜓
6	4 5 3	dvelimf	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → Ⅎ 𝑥 𝜓 )
7	6	nf5rd	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝜓 → ∀ 𝑥 𝜓 ) )