Theorem acm1s 84

Description: With is monotone in its first argument. Syllogism form of acm1 80.

Hypothesis

Ref	Expression
acm1s.1	⊦ (𝜑 ⊸ 𝜒)

Assertion

Ref	Expression
acm1s	⊦ ((𝜑 & 𝜓) ⊸ (𝜒 & 𝜓))

Proof of Theorem acm1s

Step	Hyp	Ref	Expression
1		ax-init 7	. . 3 ⊦ (~ (𝜑 & 𝜓) ⅋ (𝜑 & 𝜓))
2		acm1s.1	. . 3 ⊦ (𝜑 ⊸ 𝜒)
3	1, 2	acm1 80	. 2 ⊦ (~ (𝜑 & 𝜓) ⅋ (𝜒 & 𝜓))
4	3	dfli2i 66	1 ⊦ ((𝜑 & 𝜓) ⊸ (𝜒 & 𝜓))

Syntax hints:  ~ wneg 3   & 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:  dflb2s  92