Theorem wl-dfnan2 33296

Description: An alternative definition of "nand" based on imnan 438. See df-nan 1448 for the original definition. This theorem allows various shortenings. (Contributed by Wolf Lammen, 26-Jun-2020.)

Assertion

Ref	Expression
wl-dfnan2	⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓))

Proof of Theorem wl-dfnan2

Step	Hyp	Ref	Expression
1		df-nan 1448	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2		imnan 438	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	1, 2	bitr4i 267	1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ⊼ wnan 1447

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 197  df-an 386  df-nan 1448

This theorem is referenced by:  wl-nancom  33297  wl-nannan  33298  wl-nannot  33299  wl-nanbi1  33300  wl-nanbi2  33301