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Theorem nancom 1485
Description: Alternative denial is commutative. Remark: alternative denial is not associative, see nanass 1499. (Contributed by Mario Carneiro, 9-May-2015.) (Proof shortened by Wolf Lammen, 26-Jun-2020.)
Assertion
Ref Expression
nancom ((𝜑𝜓) ↔ (𝜓𝜑))

Proof of Theorem nancom
StepHypRef Expression
1 con2b 362 . 2 ((𝜑 → ¬ 𝜓) ↔ (𝜓 → ¬ 𝜑))
2 nanimn 1483 . 2 ((𝜑𝜓) ↔ (𝜑 → ¬ 𝜓))
3 nanimn 1483 . 2 ((𝜓𝜑) ↔ (𝜓 → ¬ 𝜑))
41, 2, 33bitr4i 305 1 ((𝜑𝜓) ↔ (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wnan 1480
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 209  df-an 399  df-nan 1481
This theorem is referenced by:  nanbi2  1491  nanass  1499  falnantru  1576
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