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Theorem oa4dcom 970
Description: Lemma commuting terms.
Assertion
Ref Expression
oa4dcom (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))) = (b ^ (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d))))))
Proof of Theorem oa4dcom
StepHypRef Expression
1 ancom 74 . . . 4 (a ^ b) = (b ^ a)
2 ancom 74 . . . 4 ((a ->1 d) ^ (b ->1 d)) = ((b ->1 d) ^ (a ->1 d))
31, 22or 72 . . 3 ((a ^ b) v ((a ->1 d) ^ (b ->1 d))) = ((b ^ a) v ((b ->1 d) ^ (a ->1 d)))
4 ancom 74 . . 3 (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))) = (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d))))
53, 42or 72 . 2 (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))) = (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))))
65lan 77 1 (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))) = (b ^ (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d))))))
Colors of variables: term
Syntax hints:   = wb 1   v 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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