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Theorem nancom 1490
Description: The 'nand' operator commutes. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 7-Mar-2020.)
Assertion
Ref Expression
nancom ((𝜑𝜓) ↔ (𝜓𝜑))

Proof of Theorem nancom
StepHypRef Expression
1 df-nan 1488 . . 3 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
2 ancom 465 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
31, 2xchbinx 323 . 2 ((𝜑𝜓) ↔ ¬ (𝜓𝜑))
4 df-nan 1488 . 2 ((𝜓𝜑) ↔ ¬ (𝜓𝜑))
53, 4bitr4i 267 1 ((𝜑𝜓) ↔ (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383  wnan 1487
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 385  df-nan 1488
This theorem is referenced by:  nanbi2  1496  falnantru  1566  rp-fakenanass  38177
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