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Theorem u1lemc4 701
Description: Lemma for Sasaki implication study.
Hypothesis
Ref Expression
ulemc3.1 a C b
Assertion
Ref Expression
u1lemc4 (a ->1 b) = (a' v b)
Proof of Theorem u1lemc4
StepHypRef Expression
1 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
2 comid 187 . . . . 5 a C a
32comcom2 183 . . . 4 a C a'
4 ulemc3.1 . . . 4 a C b
53, 4fh4 472 . . 3 (a' v (a ^ b)) = ((a' v a) ^ (a' v b))
6 ancom 74 . . . 4 ((a' v a) ^ (a' v b)) = ((a' v b) ^ (a' v a))
7 ax-a2 31 . . . . . . 7 (a' v a) = (a v a')
8 df-t 41 . . . . . . . 8 1 = (a v a')
98ax-r1 35 . . . . . . 7 (a v a') = 1
107, 9ax-r2 36 . . . . . 6 (a' v a) = 1
1110lan 77 . . . . 5 ((a' v b) ^ (a' v a)) = ((a' v b) ^ 1)
12 an1 106 . . . . 5 ((a' v b) ^ 1) = (a' v b)
1311, 12ax-r2 36 . . . 4 ((a' v b) ^ (a' v a)) = (a' v b)
146, 13ax-r2 36 . . 3 ((a' v a) ^ (a' v b)) = (a' v b)
155, 14ax-r2 36 . 2 (a' v (a ^ b)) = (a' v b)
161, 15ax-r2 36 1 (a ->1 b) = (a' v b)
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  '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:  u1lemle1  710  u1lem1  734
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