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Theorem lem3.4.6 1079
Description: Equation 3.14 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton, 3-Jul-05.)
Hypothesis
Ref Expression
lem3.4.6.1 (a5 b) = 1
Assertion
Ref Expression
lem3.4.6 ((ac) ≡5 (bc)) = 1

Proof of Theorem lem3.4.6
StepHypRef Expression
1 lem3.3.6 1056 . . . 4 (a2 (bc)) = ((ac) →2 (bc))
21ax-r1 35 . . 3 ((ac) →2 (bc)) = (a2 (bc))
3 lem3.4.6.1 . . . 4 (a5 b) = 1
43lem3.4.5 1078 . . 3 (a2 (bc)) = 1
52, 4ax-r2 36 . 2 ((ac) →2 (bc)) = 1
6 lem3.3.6 1056 . . . 4 (b2 (ac)) = ((bc) →2 (ac))
76ax-r1 35 . . 3 ((bc) →2 (ac)) = (b2 (ac))
8 df-id5 1047 . . . . 5 (b5 a) = ((ba) ∪ (ba ))
9 ancom 74 . . . . . . 7 (ba) = (ab)
10 ancom 74 . . . . . . 7 (ba ) = (ab )
119, 102or 72 . . . . . 6 ((ba) ∪ (ba )) = ((ab) ∪ (ab ))
12 df-id5 1047 . . . . . . 7 (a5 b) = ((ab) ∪ (ab ))
1312ax-r1 35 . . . . . 6 ((ab) ∪ (ab )) = (a5 b)
1411, 13, 33tr 65 . . . . 5 ((ba) ∪ (ba )) = 1
158, 14ax-r2 36 . . . 4 (b5 a) = 1
1615lem3.4.5 1078 . . 3 (b2 (ac)) = 1
177, 16ax-r2 36 . 2 ((bc) →2 (ac)) = 1
185, 17lem3.4.4 1077 1 ((ac) ≡5 (bc)) = 1
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  1wt 8  2 wi2 13  5 wid5 22
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-wom 361
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-id5 1047  df-b1 1048
This theorem is referenced by: (None)
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