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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 (c ->3 a) = (c ->3 b)
Proof of Theorem li3
StepHypRef Expression
1 li3.1 . . . . 5 a = b
21lan 77 . . . 4 (c' ^ a) = (c' ^ b)
31ax-r4 37 . . . . 5 a' = b'
43lan 77 . . . 4 (c' ^ a') = (c' ^ b')
52, 42or 72 . . 3 ((c' ^ a) v (c' ^ a')) = ((c' ^ b) v (c' ^ b'))
61lor 70 . . . 4 (c' v a) = (c' v b)
76lan 77 . . 3 (c ^ (c' v a)) = (c ^ (c' v b))
85, 72or 72 . 2 (((c' ^ a) v (c' ^ a')) v (c ^ (c' v a))) = (((c' ^ b) v (c' ^ b')) v (c ^ (c' v b)))
9 df-i3 46 . 2 (c ->3 a) = (((c' ^ a) v (c' ^ a')) v (c ^ (c' v a)))
10 df-i3 46 . 2 (c ->3 b) = (((c' ^ b) v (c' ^ b')) v (c ^ (c' v b)))
118, 9, 103tr1 63 1 (c ->3 a) = (c ->3 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v 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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