Theorem trunanfal 1611

Description: A ⊼ identity. (Contributed by Anthony Hart, 23-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)

Assertion

Ref	Expression
trunanfal	⊢ ((⊤ ⊼ ⊥) ↔ ⊤)

Proof of Theorem trunanfal

Step	Hyp	Ref	Expression
1		df-nan 1521	. . 3 ⊢ ((⊤ ⊼ ⊥) ↔ ¬ (⊤ ∧ ⊥))
2		truanfal 1603	. . 3 ⊢ ((⊤ ∧ ⊥) ↔ ⊥)
3	1, 2	xchbinx 337	. 2 ⊢ ((⊤ ⊼ ⊥) ↔ ¬ ⊥)
4		notfal 1597	. 2 ⊢ (¬ ⊥ ↔ ⊤)
5	3, 4	bitri 278	1 ⊢ ((⊤ ⊼ ⊥) ↔ ⊤)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ⊼ wnan 1520  ⊤wtru 1570  ⊥wfal 1581

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-nan 1521  df-tru 1572  df-fal 1582

This theorem is used by:  falnantru  1612