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Theorem 4oath1 1041
Description: Proper 4-OA theorem.
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
4oath1 ((a ->1 d) ^ f) = ((a ->1 d) ^ (b ->1 d))
Proof of Theorem 4oath1
StepHypRef Expression
1 4oa.1 . . . . . 6 e = (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
2 4oa.2 . . . . . 6 f = (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e)
31, 24oaiii 1040 . . . . 5 ((a ->1 d) ^ f) = ((b ->1 d) ^ f)
43lan 77 . . . 4 (((a ->1 d) ^ f) ^ ((a ->1 d) ^ f)) = (((a ->1 d) ^ f) ^ ((b ->1 d) ^ f))
5 or32 82 . . . . . . 7 (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v e) = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
62, 5ax-r2 36 . . . . . 6 f = (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))
76lan 77 . . . . 5 ((a ->1 d) ^ f) = ((a ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))))
86lan 77 . . . . 5 ((b ->1 d) ^ f) = ((b ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))))
97, 82an 79 . . . 4 (((a ->1 d) ^ f) ^ ((b ->1 d) ^ f)) = (((a ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) ^ ((b ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))))
104, 9ax-r2 36 . . 3 (((a ->1 d) ^ f) ^ ((a ->1 d) ^ f)) = (((a ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) ^ ((b ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))))
11 anidm 111 . . . 4 (((a ->1 d) ^ f) ^ ((a ->1 d) ^ f)) = ((a ->1 d) ^ f)
1211ax-r1 35 . . 3 ((a ->1 d) ^ f) = (((a ->1 d) ^ f) ^ ((a ->1 d) ^ f))
13 anandir 115 . . 3 (((a ->1 d) ^ (b ->1 d)) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) = (((a ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) ^ ((b ->1 d) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))))
1410, 12, 133tr1 63 . 2 ((a ->1 d) ^ f) = (((a ->1 d) ^ (b ->1 d)) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))))
15 ax-a2 31 . . 3 (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d))) = (((a ->1 d) ^ (b ->1 d)) v ((a ^ b) v e))
1615lan 77 . 2 (((a ->1 d) ^ (b ->1 d)) ^ (((a ^ b) v e) v ((a ->1 d) ^ (b ->1 d)))) = (((a ->1 d) ^ (b ->1 d)) ^ (((a ->1 d) ^ (b ->1 d)) v ((a ^ b) v e)))
17 anabs 121 . 2 (((a ->1 d) ^ (b ->1 d)) ^ (((a ->1 d) ^ (b ->1 d)) v ((a ^ b) v e))) = ((a ->1 d) ^ (b ->1 d))
1814, 16, 173tr 65 1 ((a ->1 d) ^ f) = ((a ->1 d) ^ (b ->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-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:  4oagen1  1042
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