Metamath Proof Explorer
Description: Equality implies 'less than or equal to'. (Contributed by Glauco
Siliprandi, 17-Aug-2020)
|
|
Ref |
Expression |
|
Hypotheses |
xreqled.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
|
|
xreqled.2 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
|
Assertion |
xreqled |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
xreqled.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
2 |
|
xreqled.2 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
3 |
|
xreqle |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵 ) → 𝐴 ≤ 𝐵 ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |