MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nancom Structured version   Unicode version

Theorem nancom 1382
Description: The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
Assertion
Ref Expression
nancom  |-  ( (
ph  -/\  ps )  <->  ( ps  -/\  ph ) )

Proof of Theorem nancom
StepHypRef Expression
1 df-nan 1380 . . 3  |-  ( (
ph  -/\  ps )  <->  -.  ( ph  /\  ps ) )
2 ancom 451 . . 3  |-  ( (
ph  /\  ps )  <->  ( ps  /\  ph )
)
31, 2xchbinx 311 . 2  |-  ( (
ph  -/\  ps )  <->  -.  ( ps  /\  ph ) )
4 df-nan 1380 . 2  |-  ( ( ps  -/\  ph )  <->  -.  ( ps  /\  ph ) )
53, 4bitr4i 255 1  |-  ( (
ph  -/\  ps )  <->  ( ps  -/\  ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    -/\ wnan 1379
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 188  df-an 372  df-nan 1380
This theorem is referenced by:  nanbi2  1392  falnantru  1484  rp-fakenanass  36073
  Copyright terms: Public domain W3C validator