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Theorem 3vded11 814
 Description: A 3-variable theorem. Experiment with weak deduction theorem.
Hypothesis
Ref Expression
3vded11.1 b ≤ (c1 (b1 a))
Assertion
Ref Expression
3vded11 c ≤ (b1 a)

Proof of Theorem 3vded11
StepHypRef Expression
1 le1 146 . . 3 (c1 (b1 a)) ≤ 1
2 df-t 41 . . . . 5 1 = ((bc ) ∪ (bc ) )
3 ancom 74 . . . . . . . 8 (cb ) = (bc)
4 anor2 89 . . . . . . . 8 (bc) = (bc )
53, 4ax-r2 36 . . . . . . 7 (cb ) = (bc )
65lor 70 . . . . . 6 ((bc ) ∪ (cb )) = ((bc ) ∪ (bc ) )
76ax-r1 35 . . . . 5 ((bc ) ∪ (bc ) ) = ((bc ) ∪ (cb ))
8 ax-a3 32 . . . . 5 ((bc ) ∪ (cb )) = (b ∪ (c ∪ (cb )))
92, 7, 83tr 65 . . . 4 1 = (b ∪ (c ∪ (cb )))
10 3vded11.1 . . . . 5 b ≤ (c1 (b1 a))
11 leo 158 . . . . . . . . 9 b ≤ (b ∪ (ba))
12 df-i1 44 . . . . . . . . . 10 (b1 a) = (b ∪ (ba))
1312ax-r1 35 . . . . . . . . 9 (b ∪ (ba)) = (b1 a)
1411, 13lbtr 139 . . . . . . . 8 b ≤ (b1 a)
1514lelan 167 . . . . . . 7 (cb ) ≤ (c ∩ (b1 a))
1615lelor 166 . . . . . 6 (c ∪ (cb )) ≤ (c ∪ (c ∩ (b1 a)))
17 df-i1 44 . . . . . . 7 (c1 (b1 a)) = (c ∪ (c ∩ (b1 a)))
1817ax-r1 35 . . . . . 6 (c ∪ (c ∩ (b1 a))) = (c1 (b1 a))
1916, 18lbtr 139 . . . . 5 (c ∪ (cb )) ≤ (c1 (b1 a))
2010, 19lel2or 170 . . . 4 (b ∪ (c ∪ (cb ))) ≤ (c1 (b1 a))
219, 20bltr 138 . . 3 1 ≤ (c1 (b1 a))
221, 21lebi 145 . 2 (c1 (b1 a)) = 1
2322u1lemle2 715 1 c ≤ (b1 a)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ 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:  3vded13  816
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