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Theorem gomaex3lem3 916
Description: Lemma for Godowski 6-var -> Mayet Example 3.
Assertion
Ref Expression
gomaex3lem3 ((p' ->1 q)' v (p' ^ q)) = p'
Proof of Theorem gomaex3lem3
StepHypRef Expression
1 anor1 88 . . . . 5 (p' ^ (p' ^ q)') = (p'' v (p' ^ q))'
21ax-r1 35 . . . 4 (p'' v (p' ^ q))' = (p' ^ (p' ^ q)')
3 df-i1 44 . . . . 5 (p' ->1 q) = (p'' v (p' ^ q))
43ax-r4 37 . . . 4 (p' ->1 q)' = (p'' v (p' ^ q))'
5 id 59 . . . 4 (p' ^ (p' ^ q)') = (p' ^ (p' ^ q)')
62, 4, 53tr1 63 . . 3 (p' ->1 q)' = (p' ^ (p' ^ q)')
76ax-r5 38 . 2 ((p' ->1 q)' v (p' ^ q)) = ((p' ^ (p' ^ q)') v (p' ^ q))
8 coman1 185 . . 3 (p' ^ q) C p'
9 comid 187 . . . 4 (p' ^ q) C (p' ^ q)
109comcom2 183 . . 3 (p' ^ q) C (p' ^ q)'
118, 10fh3r 475 . 2 ((p' ^ (p' ^ q)') v (p' ^ q)) = ((p' v (p' ^ q)) ^ ((p' ^ q)' v (p' ^ q)))
12 orabs 120 . . . 4 (p' v (p' ^ q)) = p'
13 ax-a2 31 . . . . 5 ((p' ^ q)' v (p' ^ q)) = ((p' ^ q) v (p' ^ q)')
14 df-t 41 . . . . . 6 1 = ((p' ^ q) v (p' ^ q)')
1514ax-r1 35 . . . . 5 ((p' ^ q) v (p' ^ q)') = 1
1613, 15ax-r2 36 . . . 4 ((p' ^ q)' v (p' ^ q)) = 1
1712, 162an 79 . . 3 ((p' v (p' ^ q)) ^ ((p' ^ q)' v (p' ^ q))) = (p' ^ 1)
18 an1 106 . . 3 (p' ^ 1) = p'
1917, 18ax-r2 36 . 2 ((p' v (p' ^ q)) ^ ((p' ^ q)' v (p' ^ q))) = p'
207, 11, 193tr 65 1 ((p' ->1 q)' v (p' ^ q)) = p'
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->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
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:  gomaex3lem7  920
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