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Theorem 4oaiii 1040
Description: Proper OA analog to Godowski/Greechie, Eq. III.
Hypotheses
Ref Expression
4oa.1 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
4oa.2 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
Assertion
Ref Expression
4oaiii ((a ->1 d) ^ f) = ((b ->1 d) ^ f)
Proof of Theorem 4oaiii
StepHypRef Expression
1 4oa.1 . . . 4 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
2 4oa.2 . . . 4 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
31, 24oa 1039 . . 3 ((a ->1 d) ^ f) =< (b ->1 d)
4 lear 161 . . 3 ((a ->1 d) ^ f) =< f
53, 4ler2an 173 . 2 ((a ->1 d) ^ f) =< ((b ->1 d) ^ f)
6 ancom 74 . . . . 5 (((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))))
71, 6ax-r2 36 . . . 4 e = (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d))))
8 ancom 74 . . . . . . 7 (a ^ b) = (b ^ a)
9 ancom 74 . . . . . . 7 ((a ->1 d) ^ (b ->1 d)) = ((b ->1 d) ^ (a ->1 d))
108, 92or 72 . . . . . 6 ((a ^ b) v ((a ->1 d) ^ (b ->1 d))) = ((b ^ a) v ((b ->1 d) ^ (a ->1 d)))
1110ax-r5 38 . . . . 5 (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e) = (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v e)
122, 11ax-r2 36 . . . 4 f = (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v e)
137, 124oa 1039 . . 3 ((b ->1 d) ^ f) =< (a ->1 d)
14 lear 161 . . 3 ((b ->1 d) ^ f) =< f
1513, 14ler2an 173 . 2 ((b ->1 d) ^ f) =< ((a ->1 d) ^ f)
165, 15lebi 145 1 ((a ->1 d) ^ f) = ((b ->1 d) ^ f)
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7   ->1 wi1 12
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  ax-4oa 1033
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:  4oath1  1041
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