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Theorem df2c1 468
Description: Dual 'commutes' analogue for == analogue of =.
Hypothesis
Ref Expression
df2c1.1 a = ((a v b) ^ (a v b'))
Assertion
Ref Expression
df2c1 a C b
Proof of Theorem df2c1
StepHypRef Expression
1 df2c1.1 . . . . 5 a = ((a v b) ^ (a v b'))
2 df-a 40 . . . . . 6 ((a v b) ^ (a v b')) = ((a v b)' v (a v b')')'
3 anor3 90 . . . . . . . . 9 (a' ^ b') = (a v b)'
4 anor3 90 . . . . . . . . 9 (a' ^ b'') = (a v b')'
53, 42or 72 . . . . . . . 8 ((a' ^ b') v (a' ^ b'')) = ((a v b)' v (a v b')')
65ax-r1 35 . . . . . . 7 ((a v b)' v (a v b')') = ((a' ^ b') v (a' ^ b''))
76ax-r4 37 . . . . . 6 ((a v b)' v (a v b')')' = ((a' ^ b') v (a' ^ b''))'
82, 7ax-r2 36 . . . . 5 ((a v b) ^ (a v b')) = ((a' ^ b') v (a' ^ b''))'
91, 8ax-r2 36 . . . 4 a = ((a' ^ b') v (a' ^ b''))'
109con2 67 . . 3 a' = ((a' ^ b') v (a' ^ b''))
1110df-c1 132 . 2 a' C b'
1211comcom5 458 1 a C b
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  gsth  489
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