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Theorem xorneg1 1474
Description: The connector is negated under negation of one argument. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 27-Jun-2020.)
Assertion
Ref Expression
xorneg1 ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))

Proof of Theorem xorneg1
StepHypRef Expression
1 xorcom 1466 . 2 ((¬ 𝜑𝜓) ↔ (𝜓 ⊻ ¬ 𝜑))
2 xorneg2 1473 . . 3 ((𝜓 ⊻ ¬ 𝜑) ↔ ¬ (𝜓𝜑))
3 xorcom 1466 . . 3 ((𝜓𝜑) ↔ (𝜑𝜓))
42, 3xchbinx 324 . 2 ((𝜓 ⊻ ¬ 𝜑) ↔ ¬ (𝜑𝜓))
51, 4bitri 264 1 ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wxo 1463
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-xor 1464
This theorem is referenced by:  xorneg  1475
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