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Theorem li3 252
 Description: Introduce Kalmbach implication to the left.
Hypothesis
Ref Expression
li3.1 a = b
Assertion
Ref Expression
li3 (c3 a) = (c3 b)

Proof of Theorem li3
StepHypRef Expression
1 li3.1 . . . . 5 a = b
21lan 77 . . . 4 (ca) = (cb)
31ax-r4 37 . . . . 5 a = b
43lan 77 . . . 4 (ca ) = (cb )
52, 42or 72 . . 3 ((ca) ∪ (ca )) = ((cb) ∪ (cb ))
61lor 70 . . . 4 (ca) = (cb)
76lan 77 . . 3 (c ∩ (ca)) = (c ∩ (cb))
85, 72or 72 . 2 (((ca) ∪ (ca )) ∪ (c ∩ (ca))) = (((cb) ∪ (cb )) ∪ (c ∩ (cb)))
9 df-i3 46 . 2 (c3 a) = (((ca) ∪ (ca )) ∪ (c ∩ (ca)))
10 df-i3 46 . 2 (c3 b) = (((cb) ∪ (cb )) ∪ (c ∩ (cb)))
118, 9, 103tr1 63 1 (c3 a) = (c3 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14 This theorem was proved from axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i3 46 This theorem is referenced by:  2i3  254  ud3lem0a  260  bina1  282  i31  520  i3aa  521  i3btr  528  i3li3  539  i3th2  544  i3th3  545  i3th4  546
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