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Theorem dfi4b 500
 Description: Alternate non-tollens conditional.
Assertion
Ref Expression
dfi4b (a4 b) = ((ab) ∩ ((b ∪ (ba )) ∪ (ba)))

Proof of Theorem dfi4b
StepHypRef Expression
1 i4i3 271 . 2 (a4 b) = (b3 a )
2 dfi3b 499 . . 3 (b3 a ) = ((b a ) ∩ ((b ∪ (b a )) ∪ (b a )))
3 ax-a2 31 . . . . . 6 (ab) = (ba )
4 ax-a1 30 . . . . . . 7 b = b
54ax-r5 38 . . . . . 6 (ba ) = (b a )
63, 5ax-r2 36 . . . . 5 (ab) = (b a )
74ran 78 . . . . . . . 8 (ba ) = (b a )
87lor 70 . . . . . . 7 (b ∪ (ba )) = (b ∪ (b a ))
9 ax-a1 30 . . . . . . . 8 a = a
104, 92an 79 . . . . . . 7 (ba) = (b a )
118, 102or 72 . . . . . 6 ((b ∪ (ba )) ∪ (ba)) = ((b ∪ (b a )) ∪ (b a ))
12 or32 82 . . . . . 6 ((b ∪ (b a )) ∪ (b a )) = ((b ∪ (b a )) ∪ (b a ))
1311, 12ax-r2 36 . . . . 5 ((b ∪ (ba )) ∪ (ba)) = ((b ∪ (b a )) ∪ (b a ))
146, 132an 79 . . . 4 ((ab) ∩ ((b ∪ (ba )) ∪ (ba))) = ((b a ) ∩ ((b ∪ (b a )) ∪ (b a )))
1514ax-r1 35 . . 3 ((b a ) ∩ ((b ∪ (b a )) ∪ (b a ))) = ((ab) ∩ ((b ∪ (ba )) ∪ (ba)))
162, 15ax-r2 36 . 2 (b3 a ) = ((ab) ∩ ((b ∪ (ba )) ∪ (ba)))
171, 16ax-r2 36 1 (a4 b) = ((ab) ∩ ((b ∪ (ba )) ∪ (ba)))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14   →4 wi4 15 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-i3 46  df-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  negantlem10  861
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