Theorem nems 86

Description: "Why not" is monotone.

Hypothesis

Ref	Expression
nems.1	⊦ (𝜑 ⊸ 𝜓)

Assertion

Ref	Expression
nems	⊦ (? 𝜑 ⊸ ? 𝜓)

Proof of Theorem nems

Step	Hyp	Ref	Expression
1		nems.1	. . . . 5 ⊦ (𝜑 ⊸ 𝜓)
2	1	dfli1i 65	. . . 4 ⊦ (~ 𝜑 ⅋ 𝜓)
3		ax-init 7	. . . . 5 ⊦ (~ ? 𝜓 ⅋ ? 𝜓)
4	3	ax-epe 50	. . . 4 ⊦ (~ 𝜓 ⅋ ? 𝜓)
5	2, 4	ax-cut 6	. . 3 ⊦ (~ 𝜑 ⅋ ? 𝜓)
6	5	ax-ipe 49	. 2 ⊦ (~ ? 𝜑 ⅋ ? 𝜓)
7	6	dfli2i 66	1 ⊦ (? 𝜑 ⊸ ? 𝜓)

Syntax hints:  ~ wneg 3  ? wne 48   ⊸ wli 61

This theorem was proved from axioms:  ax-ibot 4  ax-ebot 5  ax-cut 6  ax-init 7  ax-mdco 8  ax-eac1 33  ax-eac2 34  ax-ipe 49  ax-epe 50

This theorem depends on definitions:  df-lb 56  df-li 62

This theorem is referenced by: (None)