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Theorem gomaex3lem2 915
Description: Lemma for Godowski 6-var -> Mayet Example 3.
Hypothesis
Ref Expression
gomaex3lem2.5 e =< f'
Assertion
Ref Expression
gomaex3lem2 ((e v f)' v f) = e'
Proof of Theorem gomaex3lem2
StepHypRef Expression
1 gomaex3lem2.5 . . . . . 6 e =< f'
21lecon3 157 . . . . 5 f =< e'
32lecom 180 . . . 4 f C e'
4 comid 187 . . . . 5 f C f
54comcom2 183 . . . 4 f C f'
63, 5fh3r 475 . . 3 ((e' ^ f') v f) = ((e' v f) ^ (f' v f))
7 anor3 90 . . . . 5 (e' ^ f') = (e v f)'
87ax-r5 38 . . . 4 ((e' ^ f') v f) = ((e v f)' v f)
98ax-r1 35 . . 3 ((e v f)' v f) = ((e' ^ f') v f)
10 anabs 121 . . . . . 6 (e' ^ (e' v f)) = e'
1110df2le1 135 . . . . 5 e' =< (e' v f)
12 leid 148 . . . . . 6 e' =< e'
1312, 2lel2or 170 . . . . 5 (e' v f) =< e'
1411, 13lebi 145 . . . 4 e' = (e' v f)
15 df-t 41 . . . . 5 1 = (f v f')
16 ax-a2 31 . . . . 5 (f v f') = (f' v f)
1715, 16ax-r2 36 . . . 4 1 = (f' v f)
1814, 172an 79 . . 3 (e' ^ 1) = ((e' v f) ^ (f' v f))
196, 9, 183tr1 63 . 2 ((e v f)' v f) = (e' ^ 1)
20 an1 106 . 2 (e' ^ 1) = e'
2119, 20ax-r2 36 1 ((e v f)' v f) = e'
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  1wt 8
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  gomaex3lem7  920
  Copyright terms: Public domain W3C validator