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Metamath Proof Explorer

Theorem hbnae

Description: All variables are effectively bound in a distinct variable specifier. Lemma L19 in Megill p. 446 (p. 14 of the preprint). Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker hbnaev when possible. (Contributed by NM, 13-May-1993) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		hbae	⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ∀ 𝑥 𝑥 = 𝑦 )
2	1	hbn	⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ∀ 𝑧 ¬ ∀ 𝑥 𝑥 = 𝑦 )