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Theorem xordidc 1287
Description: Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.)
Assertion
Ref Expression
xordidc (DECID φ → (DECID ψ → (DECID χ → ((φ (ψχ)) ↔ ((φ ψ) ⊻ (φ χ))))))

Proof of Theorem xordidc
StepHypRef Expression
1 dcbi 843 . . . . 5 (DECID ψ → (DECID χDECID (ψχ)))
21imp 115 . . . 4 ((DECID ψ DECID χ) → DECID (ψχ))
3 annimdc 844 . . . . . 6 (DECID φ → (DECID (ψχ) → ((φ ¬ (ψχ)) ↔ ¬ (φ → (ψχ)))))
43imp 115 . . . . 5 ((DECID φ DECID (ψχ)) → ((φ ¬ (ψχ)) ↔ ¬ (φ → (ψχ))))
5 pm5.32 426 . . . . . 6 ((φ → (ψχ)) ↔ ((φ ψ) ↔ (φ χ)))
65notbii 593 . . . . 5 (¬ (φ → (ψχ)) ↔ ¬ ((φ ψ) ↔ (φ χ)))
74, 6syl6bb 185 . . . 4 ((DECID φ DECID (ψχ)) → ((φ ¬ (ψχ)) ↔ ¬ ((φ ψ) ↔ (φ χ))))
82, 7sylan2 270 . . 3 ((DECID φ (DECID ψ DECID χ)) → ((φ ¬ (ψχ)) ↔ ¬ ((φ ψ) ↔ (φ χ))))
9 xornbidc 1279 . . . . . 6 (DECID ψ → (DECID χ → ((ψχ) ↔ ¬ (ψχ))))
109imp 115 . . . . 5 ((DECID ψ DECID χ) → ((ψχ) ↔ ¬ (ψχ)))
1110adantl 262 . . . 4 ((DECID φ (DECID ψ DECID χ)) → ((ψχ) ↔ ¬ (ψχ)))
1211anbi2d 437 . . 3 ((DECID φ (DECID ψ DECID χ)) → ((φ (ψχ)) ↔ (φ ¬ (ψχ))))
13 dcan 841 . . . . . 6 (DECID φ → (DECID ψDECID (φ ψ)))
1413imp 115 . . . . 5 ((DECID φ DECID ψ) → DECID (φ ψ))
1514adantrr 448 . . . 4 ((DECID φ (DECID ψ DECID χ)) → DECID (φ ψ))
16 dcan 841 . . . . . 6 (DECID φ → (DECID χDECID (φ χ)))
1716imp 115 . . . . 5 ((DECID φ DECID χ) → DECID (φ χ))
1817adantrl 447 . . . 4 ((DECID φ (DECID ψ DECID χ)) → DECID (φ χ))
19 xornbidc 1279 . . . 4 (DECID (φ ψ) → (DECID (φ χ) → (((φ ψ) ⊻ (φ χ)) ↔ ¬ ((φ ψ) ↔ (φ χ)))))
2015, 18, 19sylc 56 . . 3 ((DECID φ (DECID ψ DECID χ)) → (((φ ψ) ⊻ (φ χ)) ↔ ¬ ((φ ψ) ↔ (φ χ))))
218, 12, 203bitr4d 209 . 2 ((DECID φ (DECID ψ DECID χ)) → ((φ (ψχ)) ↔ ((φ ψ) ⊻ (φ χ))))
2221exp32 347 1 (DECID φ → (DECID ψ → (DECID χ → ((φ (ψχ)) ↔ ((φ ψ) ⊻ (φ χ))))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  DECID wdc 741  wxo 1265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742  df-xor 1266
This theorem is referenced by: (None)
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