Theorem eqle

Description: Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005)

	Ref	Expression
Assertion	eqle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 = 𝐵 ) → 𝐴 ≤ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		leid	⊢ ( 𝐴 ∈ ℝ → 𝐴 ≤ 𝐴 )
2		breq2	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≤ 𝐴 ↔ 𝐴 ≤ 𝐵 ) )
3	2	biimpac	⊢ ( ( 𝐴 ≤ 𝐴 ∧ 𝐴 = 𝐵 ) → 𝐴 ≤ 𝐵 )
4	1 3	sylan	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 = 𝐵 ) → 𝐴 ≤ 𝐵 )