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Theorem oat 927
Description: Transformation lemma for studying the orthoarguesian law.
Hypothesis
Ref Expression
oat.1 (a' ^ (a v b)) =< c
Assertion
Ref Expression
oat b =< (a' ->1 c)
Proof of Theorem oat
StepHypRef Expression
1 leor 159 . . 3 b =< (a v b)
2 oml 445 . . . . 5 (a v (a' ^ (a v b))) = (a v b)
32ax-r1 35 . . . 4 (a v b) = (a v (a' ^ (a v b)))
4 lea 160 . . . . . 6 (a' ^ (a v b)) =< a'
5 oat.1 . . . . . 6 (a' ^ (a v b)) =< c
64, 5ler2an 173 . . . . 5 (a' ^ (a v b)) =< (a' ^ c)
76lelor 166 . . . 4 (a v (a' ^ (a v b))) =< (a v (a' ^ c))
83, 7bltr 138 . . 3 (a v b) =< (a v (a' ^ c))
91, 8letr 137 . 2 b =< (a v (a' ^ c))
10 ax-a1 30 . . . 4 a = a''
1110ax-r5 38 . . 3 (a v (a' ^ c)) = (a'' v (a' ^ c))
12 df-i1 44 . . . 4 (a' ->1 c) = (a'' v (a' ^ c))
1312ax-r1 35 . . 3 (a'' v (a' ^ c)) = (a' ->1 c)
1411, 13ax-r2 36 . 2 (a v (a' ^ c)) = (a' ->1 c)
159, 14lbtr 139 1 b =< (a' ->1 c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
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
This theorem is referenced by:  oa4ctod  968
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