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Theorem 3vded22 818
 Description: A 3-variable theorem. Experiment with weak deduction theorem.
Hypotheses
Ref Expression
3vded22.1 c ≤ ( C (a, b) ∪ C (c, b))
3vded22.2 ca
3vded22.3 c ≤ (a0 b)
Assertion
Ref Expression
3vded22 cb

Proof of Theorem 3vded22
StepHypRef Expression
1 3vded22.1 . . . 4 c ≤ ( C (a, b) ∪ C (c, b))
2 df-cmtr 134 . . . . . . 7 C (a, b) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
3 or4 84 . . . . . . 7 (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab)) ∪ ((ab ) ∪ (ab )))
42, 3ax-r2 36 . . . . . 6 C (a, b) = (((ab) ∪ (ab)) ∪ ((ab ) ∪ (ab )))
5 lear 161 . . . . . . . 8 (ab) ≤ b
6 lear 161 . . . . . . . 8 (ab) ≤ b
75, 6lel2or 170 . . . . . . 7 ((ab) ∪ (ab)) ≤ b
8 3vded22.2 . . . . . . . . . 10 ca
98lecon 154 . . . . . . . . 9 ac
109leran 153 . . . . . . . 8 (ab ) ≤ (cb )
1110lelor 166 . . . . . . 7 ((ab ) ∪ (ab )) ≤ ((ab ) ∪ (cb ))
127, 11le2or 168 . . . . . 6 (((ab) ∪ (ab)) ∪ ((ab ) ∪ (ab ))) ≤ (b ∪ ((ab ) ∪ (cb )))
134, 12bltr 138 . . . . 5 C (a, b) ≤ (b ∪ ((ab ) ∪ (cb )))
14 df-cmtr 134 . . . . . . 7 C (c, b) = (((cb) ∪ (cb )) ∪ ((cb) ∪ (cb )))
15 or4 84 . . . . . . 7 (((cb) ∪ (cb )) ∪ ((cb) ∪ (cb ))) = (((cb) ∪ (cb)) ∪ ((cb ) ∪ (cb )))
1614, 15ax-r2 36 . . . . . 6 C (c, b) = (((cb) ∪ (cb)) ∪ ((cb ) ∪ (cb )))
17 lear 161 . . . . . . . 8 (cb) ≤ b
18 lear 161 . . . . . . . 8 (cb) ≤ b
1917, 18lel2or 170 . . . . . . 7 ((cb) ∪ (cb)) ≤ b
208leran 153 . . . . . . . 8 (cb ) ≤ (ab )
2120leror 152 . . . . . . 7 ((cb ) ∪ (cb )) ≤ ((ab ) ∪ (cb ))
2219, 21le2or 168 . . . . . 6 (((cb) ∪ (cb)) ∪ ((cb ) ∪ (cb ))) ≤ (b ∪ ((ab ) ∪ (cb )))
2316, 22bltr 138 . . . . 5 C (c, b) ≤ (b ∪ ((ab ) ∪ (cb )))
2413, 23le2or 168 . . . 4 ( C (a, b) ∪ C (c, b)) ≤ ((b ∪ ((ab ) ∪ (cb ))) ∪ (b ∪ ((ab ) ∪ (cb ))))
251, 24letr 137 . . 3 c ≤ ((b ∪ ((ab ) ∪ (cb ))) ∪ (b ∪ ((ab ) ∪ (cb ))))
26 df-i0 43 . . . . 5 ((a0 b) →0 (c2 b)) = ((a0 b) ∪ (c2 b))
27 or12 80 . . . . . 6 ((ab ) ∪ (b ∪ (cb ))) = (b ∪ ((ab ) ∪ (cb )))
28 df-i0 43 . . . . . . . . 9 (a0 b) = (ab)
2928ax-r4 37 . . . . . . . 8 (a0 b) = (ab)
30 anor1 88 . . . . . . . . 9 (ab ) = (ab)
3130ax-r1 35 . . . . . . . 8 (ab) = (ab )
3229, 31ax-r2 36 . . . . . . 7 (a0 b) = (ab )
33 df-i2 45 . . . . . . 7 (c2 b) = (b ∪ (cb ))
3432, 332or 72 . . . . . 6 ((a0 b) ∪ (c2 b)) = ((ab ) ∪ (b ∪ (cb )))
35 oridm 110 . . . . . 6 ((b ∪ ((ab ) ∪ (cb ))) ∪ (b ∪ ((ab ) ∪ (cb )))) = (b ∪ ((ab ) ∪ (cb )))
3627, 34, 353tr1 63 . . . . 5 ((a0 b) ∪ (c2 b)) = ((b ∪ ((ab ) ∪ (cb ))) ∪ (b ∪ ((ab ) ∪ (cb ))))
3726, 36ax-r2 36 . . . 4 ((a0 b) →0 (c2 b)) = ((b ∪ ((ab ) ∪ (cb ))) ∪ (b ∪ ((ab ) ∪ (cb ))))
3837ax-r1 35 . . 3 ((b ∪ ((ab ) ∪ (cb ))) ∪ (b ∪ ((ab ) ∪ (cb )))) = ((a0 b) →0 (c2 b))
3925, 38lbtr 139 . 2 c ≤ ((a0 b) →0 (c2 b))
40 3vded22.3 . 2 c ≤ (a0 b)
4139, 403vded21 817 1 cb
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →2 wi2 13   C wcmtr 29 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-i0 43  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133  df-cmtr 134 This theorem is referenced by: (None)
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