Theorem gcdcomd

Description: The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024)

	Ref	Expression
Hypotheses	gcdcomd.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	gcdcomd.n	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
Assertion	gcdcomd	⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) = ( 𝑁 gcd 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		gcdcomd.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2		gcdcomd.n	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
3		gcdcom	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 gcd 𝑁 ) = ( 𝑁 gcd 𝑀 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝑀 gcd 𝑁 ) = ( 𝑁 gcd 𝑀 ) )