Theorem lmp 76

Description: Modus ponens like inference using linear implication.

Hypotheses

Ref	Expression
lmp.min	⊦ 𝜑
lmp.maj	⊦ (𝜑 ⊸ 𝜓)

Assertion

Ref	Expression
lmp	⊦ 𝜓

Proof of Theorem lmp

Step	Hyp	Ref	Expression
1		lmp.min	. . . 4 ⊦ 𝜑
2	1	ax-ibot 4	. . 3 ⊦ (⊥ ⅋ 𝜑)
3		lmp.maj	. . 3 ⊦ (𝜑 ⊸ 𝜓)
4	2, 3	mdm2i 72	. 2 ⊦ (⊥ ⅋ 𝜓)
5	4	ax-ebot 5	1 ⊦ 𝜓

Syntax hints:  ⊥wbot 1   ⊸ 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

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

This theorem is referenced by:  lbi1  89  lbi2  90  dflb  93  mcco  115