Theorem acm1 80

Description: With is monotone in its first argument.

Hypotheses

Ref	Expression
acm1.1	⊦ (𝜃 ⅋ (𝜑 & 𝜓))
acm1.2	⊦ (𝜑 ⊸ 𝜒)

Assertion

Ref	Expression
acm1	⊦ (𝜃 ⅋ (𝜒 & 𝜓))

Proof of Theorem acm1

Step	Hyp	Ref	Expression
1		acm1.1	. . . 4 ⊦ (𝜃 ⅋ (𝜑 & 𝜓))
2	1	eac1d 37	. . 3 ⊦ (𝜃 ⅋ 𝜑)
3		acm1.2	. . 3 ⊦ (𝜑 ⊸ 𝜒)
4	2, 3	mdm2i 72	. 2 ⊦ (𝜃 ⅋ 𝜒)
5	1	eac2d 39	. 2 ⊦ (𝜃 ⅋ 𝜓)
6	4, 5	iac 35	1 ⊦ (𝜃 ⅋ (𝜒 & 𝜓))

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:  acm2  81  acm1i  82  acm1s  84