Theorem syl 75

Description: Syllogism using linear implication.

Hypotheses

Ref	Expression
syl.1	⊦ (𝜑 ⊸ 𝜓)
syl.2	⊦ (𝜓 ⊸ 𝜒)

Assertion

Ref	Expression
syl	⊦ (𝜑 ⊸ 𝜒)

Proof of Theorem syl

Step	Hyp	Ref	Expression
1		syl.1	. . . 4 ⊦ (𝜑 ⊸ 𝜓)
2		df-li 62	. . . 4 ⊦ ((𝜑 ⊸ 𝜓) ⧟ (~ 𝜑 ⅋ 𝜓))
3	1, 2	lb1i 59	. . 3 ⊦ (~ 𝜑 ⅋ 𝜓)
4		syl.2	. . . 4 ⊦ (𝜓 ⊸ 𝜒)
5		df-li 62	. . . 4 ⊦ ((𝜓 ⊸ 𝜒) ⧟ (~ 𝜓 ⅋ 𝜒))
6	4, 5	lb1i 59	. . 3 ⊦ (~ 𝜓 ⅋ 𝜒)
7	3, 6	ax-cut 6	. 2 ⊦ (~ 𝜑 ⅋ 𝜒)
8		df-li 62	. 2 ⊦ ((𝜑 ⊸ 𝜒) ⧟ (~ 𝜑 ⅋ 𝜒))
9	7, 8	lb2i 60	1 ⊦ (𝜑 ⊸ 𝜒)

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-eac1 33  ax-eac2 34

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

This theorem is referenced by:  dflb2s  92  licon  94  lbeui  99  md2  114  mcco  115  abs1  178