Theorem bj-hbnaeb 33136

Description: Biconditional version of hbnae 2460 (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 2460	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)
2		sp 2201	. 2 ⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	impbii 199	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859

This theorem is referenced by: (None)