Theorem chj12i

Description: A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	chj12.1	⊢ 𝐴 ∈ Cℋ
	chj12.2	⊢ 𝐵 ∈ Cℋ
	chj12.3	⊢ 𝐶 ∈ Cℋ
Assertion	chj12i	⊢ ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		chj12.1	⊢ 𝐴 ∈ Cℋ
2		chj12.2	⊢ 𝐵 ∈ Cℋ
3		chj12.3	⊢ 𝐶 ∈ Cℋ
4	1 2	chjcomi	⊢ ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 )
5	4	oveq1i	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( ( 𝐵 ∨ℋ 𝐴 ) ∨ℋ 𝐶 )
6	1 2 3	chjassi	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) ∨ℋ 𝐶 ) = ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) )
7	2 1 3	chjassi	⊢ ( ( 𝐵 ∨ℋ 𝐴 ) ∨ℋ 𝐶 ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) )
8	5 6 7	3eqtr3i	⊢ ( 𝐴 ∨ℋ ( 𝐵 ∨ℋ 𝐶 ) ) = ( 𝐵 ∨ℋ ( 𝐴 ∨ℋ 𝐶 ) )