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Theorem df2c1 468
 Description: Dual 'commutes' analogue for ≡ analogue of =.
Hypothesis
Ref Expression
df2c1.1 a = ((ab) ∩ (ab ))
Assertion
Ref Expression
df2c1 a C b

Proof of Theorem df2c1
StepHypRef Expression
1 df2c1.1 . . . . 5 a = ((ab) ∩ (ab ))
2 df-a 40 . . . . . 6 ((ab) ∩ (ab )) = ((ab) ∪ (ab ) )
3 anor3 90 . . . . . . . . 9 (ab ) = (ab)
4 anor3 90 . . . . . . . . 9 (ab ) = (ab )
53, 42or 72 . . . . . . . 8 ((ab ) ∪ (ab )) = ((ab) ∪ (ab ) )
65ax-r1 35 . . . . . . 7 ((ab) ∪ (ab ) ) = ((ab ) ∪ (ab ))
76ax-r4 37 . . . . . 6 ((ab) ∪ (ab ) ) = ((ab ) ∪ (ab ))
82, 7ax-r2 36 . . . . 5 ((ab) ∩ (ab )) = ((ab ) ∪ (ab ))
91, 8ax-r2 36 . . . 4 a = ((ab ) ∪ (ab ))
109con2 67 . . 3 a = ((ab ) ∪ (ab ))
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   ∪ 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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