Theorem acm2 81

Description: With is monotone in its second argument.

Hypotheses

Ref	Expression
acm2.1	⊦ (𝜃 ⅋ (𝜑 & 𝜓))
acm2.2	⊦ (𝜓 ⊸ 𝜒)

Assertion

Ref	Expression
acm2	⊦ (𝜃 ⅋ (𝜑 & 𝜒))

Proof of Theorem acm2

Step	Hyp	Ref	Expression
1		acm2.1	. . . 4 ⊦ (𝜃 ⅋ (𝜑 & 𝜓))
2	1	acco 43	. . 3 ⊦ (𝜃 ⅋ (𝜓 & 𝜑))
3		acm2.2	. . 3 ⊦ (𝜓 ⊸ 𝜒)
4	2, 3	acm1 80	. 2 ⊦ (𝜃 ⅋ (𝜒 & 𝜑))
5	4	acco 43	1 ⊦ (𝜃 ⅋ (𝜑 & 𝜒))

Syntax hints:   ⅋ wmd 2   & wac 30   ⊸ 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-iac 32  ax-eac1 33  ax-eac2 34

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

This theorem is referenced by:  acm2i  83  acm2s  85