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Theorem licon 94
Description: Contrapositive rule for linear implication. This follows quite neatly from df-li 62.
Assertion
Ref Expression
licon ((𝜑𝜓) ⊸ (~ 𝜓 ⊸ ~ 𝜑))

Proof of Theorem licon
StepHypRef Expression
1 id 77 . . 3 ((𝜑𝜓) ⊸ (𝜑𝜓))
2 df-li 62 . . 3 ((𝜑𝜓) ⧟ (~ 𝜑𝜓))
31, 2lb1s 67 . 2 ((𝜑𝜓) ⊸ (~ 𝜑𝜓))
4 dnis 78 . . . . . . 7 (𝜓 ⊸ ~ ~ 𝜓)
54mdm2s 74 . . . . . 6 ((~ 𝜑𝜓) ⊸ (~ 𝜑 ⅋ ~ ~ 𝜓))
65dfli1i 65 . . . . 5 (~ (~ 𝜑𝜓) ⅋ (~ 𝜑 ⅋ ~ ~ 𝜓))
76mdcod 11 . . . 4 (~ (~ 𝜑𝜓) ⅋ (~ ~ 𝜓 ⅋ ~ 𝜑))
87dfli2i 66 . . 3 ((~ 𝜑𝜓) ⊸ (~ ~ 𝜓 ⅋ ~ 𝜑))
9 df-li 62 . . 3 ((~ 𝜓 ⊸ ~ 𝜑) ⧟ (~ ~ 𝜓 ⅋ ~ 𝜑))
108, 9lb2s 68 . 2 ((~ 𝜑𝜓) ⊸ (~ 𝜓 ⊸ ~ 𝜑))
113, 10syl 75 1 ((𝜑𝜓) ⊸ (~ 𝜓 ⊸ ~ 𝜑))
Colors of variables: wff var nilad
Syntax hints:  wmd 2  ~ wneg 3  wli 61
This theorem was proved from axioms:  ax-ibot 4  ax-ebot 5  ax-cut 6  ax-init 7  ax-mdco 8  ax-mdas 9  ax-eac1 33  ax-eac2 34
This theorem depends on definitions:  df-lb 56  df-li 62
This theorem is referenced by:  mcco  115
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