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Theorem 4oagen1b 1043
Description: "Generalized" OA.
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)
4oagen1b.1 g =< f
4oagen1b.2 h =< (a ->1 d)
Assertion
Ref Expression
4oagen1b (h ^ (g v ((a ->1 d) ^ (b ->1 d)))) = (h ^ (b ->1 d))
Proof of Theorem 4oagen1b
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)
3 4oagen1b.1 . . . 4 g =< f
41, 2, 34oagen1 1042 . . 3 ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))) = ((a ->1 d) ^ (b ->1 d))
54lan 77 . 2 (h ^ ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d))))) = (h ^ ((a ->1 d) ^ (b ->1 d)))
6 anass 76 . . . 4 ((h ^ (a ->1 d)) ^ (g v ((a ->1 d) ^ (b ->1 d)))) = (h ^ ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d)))))
76ax-r1 35 . . 3 (h ^ ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d))))) = ((h ^ (a ->1 d)) ^ (g v ((a ->1 d) ^ (b ->1 d))))
8 4oagen1b.2 . . . . 5 h =< (a ->1 d)
98df2le2 136 . . . 4 (h ^ (a ->1 d)) = h
109ran 78 . . 3 ((h ^ (a ->1 d)) ^ (g v ((a ->1 d) ^ (b ->1 d)))) = (h ^ (g v ((a ->1 d) ^ (b ->1 d))))
117, 10ax-r2 36 . 2 (h ^ ((a ->1 d) ^ (g v ((a ->1 d) ^ (b ->1 d))))) = (h ^ (g v ((a ->1 d) ^ (b ->1 d))))
12 anass 76 . . . 4 ((h ^ (a ->1 d)) ^ (b ->1 d)) = (h ^ ((a ->1 d) ^ (b ->1 d)))
1312ax-r1 35 . . 3 (h ^ ((a ->1 d) ^ (b ->1 d))) = ((h ^ (a ->1 d)) ^ (b ->1 d))
149ran 78 . . 3 ((h ^ (a ->1 d)) ^ (b ->1 d)) = (h ^ (b ->1 d))
1513, 14ax-r2 36 . 2 (h ^ ((a ->1 d) ^ (b ->1 d))) = (h ^ (b ->1 d))
165, 11, 153tr2 64 1 (h ^ (g v ((a ->1 d) ^ (b ->1 d)))) = (h ^ (b ->1 d))
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2   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:  4oadist  1044
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