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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 DECID DECID  \/_  \/_

Proof of Theorem xordidc
StepHypRef Expression
1 dcbi 843 . . . . 5 DECID DECID DECID
21imp 115 . . . 4 DECID DECID DECID
3 annimdc 844 . . . . . 6 DECID DECID
43imp 115 . . . . 5 DECID DECID
5 pm5.32 426 . . . . . 6
65notbii 593 . . . . 5
74, 6syl6bb 185 . . . 4 DECID DECID
82, 7sylan2 270 . . 3 DECID DECID DECID
9 xornbidc 1279 . . . . . 6 DECID DECID 
\/_
109imp 115 . . . . 5 DECID DECID  \/_
1110adantl 262 . . . 4 DECID DECID DECID  \/_
1211anbi2d 437 . . 3 DECID DECID DECID 
\/_
13 dcan 841 . . . . . 6 DECID DECID DECID
1413imp 115 . . . . 5 DECID DECID DECID
1514adantrr 448 . . . 4 DECID DECID DECID DECID
16 dcan 841 . . . . . 6 DECID DECID DECID
1716imp 115 . . . . 5 DECID DECID DECID
1817adantrl 447 . . . 4 DECID DECID DECID DECID
19 xornbidc 1279 . . . 4 DECID DECID  \/_
2015, 18, 19sylc 56 . . 3 DECID DECID DECID  \/_
218, 12, 203bitr4d 209 . 2 DECID DECID DECID 
\/_  \/_
2221exp32 347 1 DECID DECID DECID  \/_  \/_
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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