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Metamath Proof Explorer

Theorem hbnaes

Description: Rule that applies hbnae to antecedent. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 15-May-1993) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	hbnaes.1	⊢ ( ∀ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )
	Assertion	hbnaes	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		hbnaes.1	⊢ ( ∀ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )
2		hbnae	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦 )
3	2 1	syl	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → 𝜑 )