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Theorem fh3r 475
 Description: Foulis-Holland Theorem.
Hypotheses
Ref Expression
fh.1 a C b
fh.2 a C c
Assertion
Ref Expression
fh3r ((bc) ∪ a) = ((ba) ∩ (ca))

Proof of Theorem fh3r
StepHypRef Expression
1 fh.1 . . 3 a C b
2 fh.2 . . 3 a C c
31, 2fh3 471 . 2 (a ∪ (bc)) = ((ab) ∩ (ac))
4 ax-a2 31 . 2 ((bc) ∪ a) = (a ∪ (bc))
5 ax-a2 31 . . 3 (ba) = (ab)
6 ax-a2 31 . . 3 (ca) = (ac)
75, 62an 79 . 2 ((ba) ∩ (ca)) = ((ab) ∩ (ac))
83, 4, 73tr1 63 1 ((bc) ∪ a) = ((ba) ∩ (ca))
 Colors of variables: term Syntax hints:   = wb 1   C wc 3   ∪ wo 6   ∩ 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-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  fh3rc  481  ud1lem1  560  ud4lem2  582  ud4lem3b  584  ud5lem3  594  u2lembi  721  u4lem6  768  u1lem11  780  u3lem13b  790  mhlem  876  gomaex3lem2  915  gomaex3lem3  916
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