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Theorem oa4dcom 970
 Description: Lemma commuting terms.
Assertion
Ref Expression
oa4dcom (b ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))) = (b ∩ (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d))))))

Proof of Theorem oa4dcom
StepHypRef Expression
1 ancom 74 . . . 4 (ab) = (ba)
2 ancom 74 . . . 4 ((a1 d) ∩ (b1 d)) = ((b1 d) ∩ (a1 d))
31, 22or 72 . . 3 ((ab) ∪ ((a1 d) ∩ (b1 d))) = ((ba) ∪ ((b1 d) ∩ (a1 d)))
4 ancom 74 . . 3 (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))) = (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d))))
53, 42or 72 . 2 (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))) = (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))))
65lan 77 1 (b ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))) = (b ∩ (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d))))))
 Colors of variables: term Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7   →1 wi1 12 This theorem was proved from axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40 This theorem is referenced by:  axoa4d  1038
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