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Theorem ud5lem0c 281
Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud5lem0c (a ->5 b)' = (((a' v b') ^ (a v b')) ^ (a v b))
Proof of Theorem ud5lem0c
StepHypRef Expression
1 df-i5 48 . . 3 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
2 oran 87 . . . 4 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (((a ^ b) v (a' ^ b))' ^ (a' ^ b')')'
3 oran 87 . . . . . . . 8 ((a ^ b) v (a' ^ b)) = ((a ^ b)' ^ (a' ^ b)')'
4 df-a 40 . . . . . . . . . . 11 (a ^ b) = (a' v b')'
54con2 67 . . . . . . . . . 10 (a ^ b)' = (a' v b')
6 anor2 89 . . . . . . . . . . 11 (a' ^ b) = (a v b')'
76con2 67 . . . . . . . . . 10 (a' ^ b)' = (a v b')
85, 72an 79 . . . . . . . . 9 ((a ^ b)' ^ (a' ^ b)') = ((a' v b') ^ (a v b'))
98ax-r4 37 . . . . . . . 8 ((a ^ b)' ^ (a' ^ b)')' = ((a' v b') ^ (a v b'))'
103, 9ax-r2 36 . . . . . . 7 ((a ^ b) v (a' ^ b)) = ((a' v b') ^ (a v b'))'
1110con2 67 . . . . . 6 ((a ^ b) v (a' ^ b))' = ((a' v b') ^ (a v b'))
12 oran 87 . . . . . . 7 (a v b) = (a' ^ b')'
1312ax-r1 35 . . . . . 6 (a' ^ b')' = (a v b)
1411, 132an 79 . . . . 5 (((a ^ b) v (a' ^ b))' ^ (a' ^ b')') = (((a' v b') ^ (a v b')) ^ (a v b))
1514ax-r4 37 . . . 4 (((a ^ b) v (a' ^ b))' ^ (a' ^ b')')' = (((a' v b') ^ (a v b')) ^ (a v b))'
162, 15ax-r2 36 . . 3 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (((a' v b') ^ (a v b')) ^ (a v b))'
171, 16ax-r2 36 . 2 (a ->5 b) = (((a' v b') ^ (a v b')) ^ (a v b))'
1817con2 67 1 (a ->5 b)' = (((a' v b') ^ (a v b')) ^ (a v b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->5 wi5 16
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i5 48
This theorem is referenced by:  ud5lem1b  587  ud5lem1c  588  ud5lem3b  592  ud5lem3c  593
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