Theorem trubifal 1347

Description: A ↔ identity. (Contributed by David A. Wheeler, 23-Feb-2018.)

Assertion

Ref	Expression
trubifal	⊢ ((⊤ ↔ ⊥) ↔ ⊥)

Proof of Theorem trubifal

Step	Hyp	Ref	Expression
1		dfbi2 380	. 2 ⊢ ((⊤ ↔ ⊥) ↔ ((⊤ → ⊥) ∧ (⊥ → ⊤)))
2		truimfal 1341	. . 3 ⊢ ((⊤ → ⊥) ↔ ⊥)
3		falimtru 1342	. . 3 ⊢ ((⊥ → ⊤) ↔ ⊤)
4	2, 3	anbi12i 447	. 2 ⊢ (((⊤ → ⊥) ∧ (⊥ → ⊤)) ↔ (⊥ ∧ ⊤))
5		falantru 1334	. 2 ⊢ ((⊥ ∧ ⊤) ↔ ⊥)
6	1, 4, 5	3bitri 204	1 ⊢ ((⊤ ↔ ⊥) ↔ ⊥)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ⊤wtru 1285  ⊥wfal 1289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  falbitru  1348