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Theorem i3orlem7 558
Description: Lemma for Kalmbach implication OR builder.
Assertion
Ref Expression
i3orlem7 (a ^ b') =< ((a ->3 b)' v ((a v c) ->3 (b v c)))
Proof of Theorem i3orlem7
StepHypRef Expression
1 lea 160 . . . . . . 7 (a ^ b') =< a
2 leo 158 . . . . . . 7 a =< (a v b)
31, 2letr 137 . . . . . 6 (a ^ b') =< (a v b)
4 leo 158 . . . . . 6 (a ^ b') =< ((a ^ b') v (a ^ b))
53, 4ler2an 173 . . . . 5 (a ^ b') =< ((a v b) ^ ((a ^ b') v (a ^ b)))
65ler 149 . . . 4 (a ^ b') =< (((a v b) ^ ((a ^ b') v (a ^ b))) v ((a v b) ^ (a' ^ (a v b'))))
7 i3n1 249 . . . . . . 7 (a' ->3 b') = (((a ^ b') v (a ^ b)) v (a' ^ (a v b')))
87lan 77 . . . . . 6 ((a v b) ^ (a' ->3 b')) = ((a v b) ^ (((a ^ b') v (a ^ b)) v (a' ^ (a v b'))))
9 comor1 461 . . . . . . . . 9 (a v b) C a
10 comor2 462 . . . . . . . . . 10 (a v b) C b
1110comcom2 183 . . . . . . . . 9 (a v b) C b'
129, 11com2an 484 . . . . . . . 8 (a v b) C (a ^ b')
139, 10com2an 484 . . . . . . . 8 (a v b) C (a ^ b)
1412, 13com2or 483 . . . . . . 7 (a v b) C ((a ^ b') v (a ^ b))
159comcom2 183 . . . . . . . 8 (a v b) C a'
169, 11com2or 483 . . . . . . . 8 (a v b) C (a v b')
1715, 16com2an 484 . . . . . . 7 (a v b) C (a' ^ (a v b'))
1814, 17fh1 469 . . . . . 6 ((a v b) ^ (((a ^ b') v (a ^ b)) v (a' ^ (a v b')))) = (((a v b) ^ ((a ^ b') v (a ^ b))) v ((a v b) ^ (a' ^ (a v b'))))
198, 18ax-r2 36 . . . . 5 ((a v b) ^ (a' ->3 b')) = (((a v b) ^ ((a ^ b') v (a ^ b))) v ((a v b) ^ (a' ^ (a v b'))))
2019ax-r1 35 . . . 4 (((a v b) ^ ((a ^ b') v (a ^ b))) v ((a v b) ^ (a' ^ (a v b')))) = ((a v b) ^ (a' ->3 b'))
216, 20lbtr 139 . . 3 (a ^ b') =< ((a v b) ^ (a' ->3 b'))
2221ler 149 . 2 (a ^ b') =< (((a v b) ^ (a' ->3 b')) v ((a v c) ->3 (b v c)))
23 i3orlem6 557 . . 3 ((a ->3 b)' v ((a v c) ->3 (b v c))) = (((a v b) ^ (a' ->3 b')) v ((a v c) ->3 (b v c)))
2423ax-r1 35 . 2 (((a v b) ^ (a' ->3 b')) v ((a v c) ->3 (b v c))) = ((a ->3 b)' v ((a v c) ->3 (b v c)))
2522, 24lbtr 139 1 (a ^ b') =< ((a ->3 b)' v ((a v c) ->3 (b v c)))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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