Theorem nadd32d

Description: Commutative/associative law that swaps the last two terms in a triple sum. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026)

	Ref	Expression
Hypotheses	nadd.1	⊢ ( 𝜑 → 𝐴 ∈ On )
	nadd.2	⊢ ( 𝜑 → 𝐵 ∈ On )
	nadd.3	⊢ ( 𝜑 → 𝐶 ∈ On )
Assertion	nadd32d	⊢ ( 𝜑 → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ( ( 𝐴 +no 𝐶 ) +no 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		nadd.1	⊢ ( 𝜑 → 𝐴 ∈ On )
2		nadd.2	⊢ ( 𝜑 → 𝐵 ∈ On )
3		nadd.3	⊢ ( 𝜑 → 𝐶 ∈ On )
4		nadd32	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ( ( 𝐴 +no 𝐶 ) +no 𝐵 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ( ( 𝐴 +no 𝐶 ) +no 𝐵 ) )