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Theorem wwfh4 219
 Description: Foulis-Holland Theorem (weak).
Hypotheses
Ref Expression
wwfh4.1 a C b
wwfh4.2 c C a
Assertion
Ref Expression
wwfh4 ((b ∪ (ac)) ≡ ((ba) ∩ (bc))) = 1

Proof of Theorem wwfh4
StepHypRef Expression
1 conb 122 . . 3 ((b ∪ (ac)) ≡ ((ba) ∩ (bc))) = ((b ∪ (ac)) ≡ ((ba) ∩ (bc)) )
2 oran 87 . . . . . 6 (b ∪ (ac)) = (b ∩ (ac) )
3 df-a 40 . . . . . . . . 9 (ac) = (ac )
43con2 67 . . . . . . . 8 (ac) = (ac )
54lan 77 . . . . . . 7 (b ∩ (ac) ) = (b ∩ (ac ))
65ax-r4 37 . . . . . 6 (b ∩ (ac) ) = (b ∩ (ac ))
72, 6ax-r2 36 . . . . 5 (b ∪ (ac)) = (b ∩ (ac ))
87con2 67 . . . 4 (b ∪ (ac)) = (b ∩ (ac ))
9 df-a 40 . . . . . 6 ((ba) ∩ (bc)) = ((ba) ∪ (bc) )
10 oran 87 . . . . . . . . 9 (ba) = (ba )
1110con2 67 . . . . . . . 8 (ba) = (ba )
12 oran 87 . . . . . . . . 9 (bc) = (bc )
1312con2 67 . . . . . . . 8 (bc) = (bc )
1411, 132or 72 . . . . . . 7 ((ba) ∪ (bc) ) = ((ba ) ∪ (bc ))
1514ax-r4 37 . . . . . 6 ((ba) ∪ (bc) ) = ((ba ) ∪ (bc ))
169, 15ax-r2 36 . . . . 5 ((ba) ∩ (bc)) = ((ba ) ∪ (bc ))
1716con2 67 . . . 4 ((ba) ∩ (bc)) = ((ba ) ∪ (bc ))
188, 172bi 99 . . 3 ((b ∪ (ac)) ≡ ((ba) ∩ (bc)) ) = ((b ∩ (ac )) ≡ ((ba ) ∪ (bc )))
191, 18ax-r2 36 . 2 ((b ∪ (ac)) ≡ ((ba) ∩ (bc))) = ((b ∩ (ac )) ≡ ((ba ) ∪ (bc )))
20 wwfh4.1 . . . 4 a C b
2120comcom2 183 . . 3 a C b
22 ax-a1 30 . . . . . 6 c = c
2322ax-r1 35 . . . . 5 c = c
24 wwfh4.2 . . . . 5 c C a
2523, 24bctr 181 . . . 4 c C a
2625comcom2 183 . . 3 c C a
2721, 26wwfh2 217 . 2 ((b ∩ (ac )) ≡ ((ba ) ∪ (bc ))) = 1
2819, 27ax-r2 36 1 ((b ∪ (ac)) ≡ ((ba) ∩ (bc))) = 1
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8 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 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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