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Theorem xorass 1308
 Description: ⊻ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
xorass (((φψ) ⊻ χ) ↔ (φ ⊻ (ψχ)))

Proof of Theorem xorass
StepHypRef Expression
1 biass 348 . . . . . 6 (((φψ) ↔ χ) ↔ (φ ↔ (ψχ)))
21notbii 287 . . . . 5 (¬ ((φψ) ↔ χ) ↔ ¬ (φ ↔ (ψχ)))
3 nbbn 347 . . . . 5 ((¬ (φψ) ↔ χ) ↔ ¬ ((φψ) ↔ χ))
4 pm5.18 345 . . . . . 6 ((φ ↔ (ψχ)) ↔ ¬ (φ ↔ ¬ (ψχ)))
54con2bii 322 . . . . 5 ((φ ↔ ¬ (ψχ)) ↔ ¬ (φ ↔ (ψχ)))
62, 3, 53bitr4i 268 . . . 4 ((¬ (φψ) ↔ χ) ↔ (φ ↔ ¬ (ψχ)))
7 df-xor 1305 . . . . 5 ((φψ) ↔ ¬ (φψ))
87bibi1i 305 . . . 4 (((φψ) ↔ χ) ↔ (¬ (φψ) ↔ χ))
9 df-xor 1305 . . . . 5 ((ψχ) ↔ ¬ (ψχ))
109bibi2i 304 . . . 4 ((φ ↔ (ψχ)) ↔ (φ ↔ ¬ (ψχ)))
116, 8, 103bitr4i 268 . . 3 (((φψ) ↔ χ) ↔ (φ ↔ (ψχ)))
1211notbii 287 . 2 (¬ ((φψ) ↔ χ) ↔ ¬ (φ ↔ (ψχ)))
13 df-xor 1305 . 2 (((φψ) ⊻ χ) ↔ ¬ ((φψ) ↔ χ))
14 df-xor 1305 . 2 ((φ ⊻ (ψχ)) ↔ ¬ (φ ↔ (ψχ)))
1512, 13, 143bitr4i 268 1 (((φψ) ⊻ χ) ↔ (φ ⊻ (ψχ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ⊻ wxo 1304 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 177  df-xor 1305 This theorem is referenced by:  hadass  1386
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