Theorem naddassd

Description: Natural addition associates. 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	naddassd	⊢ ( 𝜑 → ( ( 𝐴 +no 𝐵 ) +no 𝐶 ) = ( 𝐴 +no ( 𝐵 +no 𝐶 ) ) )

Proof

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