Metamath Proof Explorer


Theorem nancom

Description: Alternative denial is commutative. Remark: alternative denial is not associative, see nanass . (Contributed by Mario Carneiro, 9-May-2015) (Proof shortened by Wolf Lammen, 26-Jun-2020)

Ref Expression
Assertion nancom
|- ( ( ph -/\ ps ) <-> ( ps -/\ ph ) )

Proof

Step Hyp Ref Expression
1 con2b
 |-  ( ( ph -> -. ps ) <-> ( ps -> -. ph ) )
2 dfnan2
 |-  ( ( ph -/\ ps ) <-> ( ph -> -. ps ) )
3 dfnan2
 |-  ( ( ps -/\ ph ) <-> ( ps -> -. ph ) )
4 1 2 3 3bitr4i
 |-  ( ( ph -/\ ps ) <-> ( ps -/\ ph ) )