Theorem licon 94

Description: Contrapositive rule for linear implication. This follows quite neatly from df-li 62.

Assertion

Ref	Expression
licon	⊦ ((𝜑 ⊸ 𝜓) ⊸ (~ 𝜓 ⊸ ~ 𝜑))

Proof of Theorem licon

Step	Hyp	Ref	Expression
1		id 77	. . 3 ⊦ ((𝜑 ⊸ 𝜓) ⊸ (𝜑 ⊸ 𝜓))
2		df-li 62	. . 3 ⊦ ((𝜑 ⊸ 𝜓) ⧟ (~ 𝜑 ⅋ 𝜓))
3	1, 2	lb1s 67	. 2 ⊦ ((𝜑 ⊸ 𝜓) ⊸ (~ 𝜑 ⅋ 𝜓))
4		dnis 78	. . . . . . 7 ⊦ (𝜓 ⊸ ~ ~ 𝜓)
5	4	mdm2s 74	. . . . . 6 ⊦ ((~ 𝜑 ⅋ 𝜓) ⊸ (~ 𝜑 ⅋ ~ ~ 𝜓))
6	5	dfli1i 65	. . . . 5 ⊦ (~ (~ 𝜑 ⅋ 𝜓) ⅋ (~ 𝜑 ⅋ ~ ~ 𝜓))
7	6	mdcod 11	. . . 4 ⊦ (~ (~ 𝜑 ⅋ 𝜓) ⅋ (~ ~ 𝜓 ⅋ ~ 𝜑))
8	7	dfli2i 66	. . 3 ⊦ ((~ 𝜑 ⅋ 𝜓) ⊸ (~ ~ 𝜓 ⅋ ~ 𝜑))
9		df-li 62	. . 3 ⊦ ((~ 𝜓 ⊸ ~ 𝜑) ⧟ (~ ~ 𝜓 ⅋ ~ 𝜑))
10	8, 9	lb2s 68	. 2 ⊦ ((~ 𝜑 ⅋ 𝜓) ⊸ (~ 𝜓 ⊸ ~ 𝜑))
11	3, 10	syl 75	1 ⊦ ((𝜑 ⊸ 𝜓) ⊸ (~ 𝜓 ⊸ ~ 𝜑))

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