Theorem acas 44

Description: & is associative.

Hypothesis

Ref	Expression
acas.1	⊦ (𝜑 ⅋ ((𝜓 & 𝜒) & 𝜃))

Assertion

Ref	Expression
acas	⊦ (𝜑 ⅋ (𝜓 & (𝜒 & 𝜃)))

Proof of Theorem acas

Step	Hyp	Ref	Expression
1		acas.1	. . . 4 ⊦ (𝜑 ⅋ ((𝜓 & 𝜒) & 𝜃))
2	1	eac1d 37	. . 3 ⊦ (𝜑 ⅋ (𝜓 & 𝜒))
3	2	eac1d 37	. 2 ⊦ (𝜑 ⅋ 𝜓)
4	2	eac2d 39	. . 3 ⊦ (𝜑 ⅋ 𝜒)
5	1	eac2d 39	. . 3 ⊦ (𝜑 ⅋ 𝜃)
6	4, 5	iac 35	. 2 ⊦ (𝜑 ⅋ (𝜒 & 𝜃))
7	3, 6	iac 35	1 ⊦ (𝜑 ⅋ (𝜓 & (𝜒 & 𝜃)))

Syntax hints:   ⅋ wmd 2   & wac 30

This theorem was proved from axioms:  ax-ibot 4  ax-ebot 5  ax-cut 6  ax-init 7  ax-mdco 8  ax-iac 32  ax-eac1 33  ax-eac2 34

This theorem is referenced by: (None)