Theorem naddcomd

Description: Natural addition commutes. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026)

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

Proof

Step	Hyp	Ref	Expression
1		nadd.1	⊢ ( 𝜑 → 𝐴 ∈ On )
2		nadd.2	⊢ ( 𝜑 → 𝐵 ∈ On )
3		naddcom	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +no 𝐵 ) = ( 𝐵 +no 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 +no 𝐵 ) = ( 𝐵 +no 𝐴 ) )