Theorem bj-hbnaeb 37310

Description: Biconditional version of hbnae 2465 (to replace it?). (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-hbnaeb	⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Proof of Theorem bj-hbnaeb

Step	Hyp	Ref	Expression
1		hbnae 2465	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2		sp 2220	. 2 ⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	impbii 211	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-11 2193  ax-12 2214  ax-13 2405

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806

This theorem is referenced by: (None)