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Theorem xorass 1637
Description: The connector is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 20-Jun-2020.)
Assertion
Ref Expression
xorass (((𝜑𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))

Proof of Theorem xorass
StepHypRef Expression
1 xor3 373 . . 3 (¬ (𝜑 ↔ (𝜓𝜒)) ↔ (𝜑 ↔ ¬ (𝜓𝜒)))
2 biass 375 . . . 4 (((𝜑𝜓) ↔ 𝜒) ↔ (𝜑 ↔ (𝜓𝜒)))
3 xnor 1635 . . . . 5 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
43bibi1i 329 . . . 4 (((𝜑𝜓) ↔ 𝜒) ↔ (¬ (𝜑𝜓) ↔ 𝜒))
5 xnor 1635 . . . . 5 ((𝜓𝜒) ↔ ¬ (𝜓𝜒))
65bibi2i 328 . . . 4 ((𝜑 ↔ (𝜓𝜒)) ↔ (𝜑 ↔ ¬ (𝜓𝜒)))
72, 4, 63bitr3i 292 . . 3 ((¬ (𝜑𝜓) ↔ 𝜒) ↔ (𝜑 ↔ ¬ (𝜓𝜒)))
8 nbbn 374 . . 3 ((¬ (𝜑𝜓) ↔ 𝜒) ↔ ¬ ((𝜑𝜓) ↔ 𝜒))
91, 7, 83bitr2ri 291 . 2 (¬ ((𝜑𝜓) ↔ 𝜒) ↔ ¬ (𝜑 ↔ (𝜓𝜒)))
10 df-xor 1634 . 2 (((𝜑𝜓) ⊻ 𝜒) ↔ ¬ ((𝜑𝜓) ↔ 𝜒))
11 df-xor 1634 . 2 ((𝜑 ⊻ (𝜓𝜒)) ↔ ¬ (𝜑 ↔ (𝜓𝜒)))
129, 10, 113bitr4i 294 1 (((𝜑𝜓) ⊻ 𝜒) ↔ (𝜑 ⊻ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wxo 1633
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 198  df-xor 1634
This theorem is referenced by:  hadass  1706  symdifass  4016
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