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Theorem fh1rc 479
Description: Foulis-Holland Theorem. (Contributed by NM, 10-Mar-2002.)
Hypotheses
Ref Expression
fh.1 a C b
fh.2 a C c
Assertion
Ref Expression
fh1rc ((cb) ∩ a) = ((ca) ∪ (ba))

Proof of Theorem fh1rc
StepHypRef Expression
1 fh.1 . . 3 a C b
2 fh.2 . . 3 a C c
31, 2fh1r 473 . 2 ((bc) ∩ a) = ((ba) ∪ (ca))
4 ax-a2 31 . . 3 (cb) = (bc)
54ran 78 . 2 ((cb) ∩ a) = ((bc) ∩ a)
6 ax-a2 31 . 2 ((ca) ∪ (ba)) = ((ba) ∪ (ca))
73, 5, 63tr1 63 1 ((cb) ∩ a) = ((ca) ∪ (ba))
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:  mhlemlem1  874
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