Theorem ablcomd

Description: An abelian group operation is commutative, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026)

	Ref	Expression
Hypotheses	ablcomd.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	ablcomd.2	⊢ + = ( +g ‘ 𝐺 )
	ablcomd.3	⊢ ( 𝜑 → 𝐺 ∈ Abel )
	ablcomd.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ablcomd.5	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion	ablcomd	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		ablcomd.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		ablcomd.2	⊢ + = ( +g ‘ 𝐺 )
3		ablcomd.3	⊢ ( 𝜑 → 𝐺 ∈ Abel )
4		ablcomd.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		ablcomd.5	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6	1 2	ablcom	⊢ ( ( 𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
7	3 4 5 6	syl3anc	⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )