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Theorem comanbn 873
Description: Biconditional commutation law.
Assertion
Ref Expression
comanbn (a' ^ b') C ((a == c) ^ (b == c))
Proof of Theorem comanbn
StepHypRef Expression
1 comanb 872 . 2 (a' ^ b') C ((a' == c') ^ (b' == c'))
2 conb 122 . . . 4 (a == c) = (a' == c')
3 conb 122 . . . 4 (b == c) = (b' == c')
42, 32an 79 . . 3 ((a == c) ^ (b == c)) = ((a' == c') ^ (b' == c'))
54ax-r1 35 . 2 ((a' == c') ^ (b' == c')) = ((a == c) ^ (b == c))
61, 5cbtr 182 1 (a' ^ b') C ((a == c) ^ (b == c))
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   == tb 5   ^ wa 7
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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