Theorem nancom 1290

Description: The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.)

Assertion

Ref	Expression
nancom	⊢ ((φ ⊼ ψ) ↔ (ψ ⊼ φ))

Proof of Theorem nancom

Step	Hyp	Ref	Expression
1		ancom 437	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
2	1	notbii 287	. 2 ⊢ (¬ (φ ∧ ψ) ↔ ¬ (ψ ∧ φ))
3		df-nan 1288	. 2 ⊢ ((φ ⊼ ψ) ↔ ¬ (φ ∧ ψ))
4		df-nan 1288	. 2 ⊢ ((ψ ⊼ φ) ↔ ¬ (ψ ∧ φ))
5	2, 3, 4	3bitr4i 268	1 ⊢ ((φ ⊼ ψ) ↔ (ψ ⊼ φ))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-nan 1288

This theorem is referenced by:  nanbi2  1296  falnantru  1356  nincom  3226