Theorem leeq1d

Description: Specialization of breq1d to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	leeq1d.1	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
	leeq1d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	leeq1d.3	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	leeq1d.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion	leeq1d	⊢ ( 𝜑 → 𝐵 ≤ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		leeq1d.1	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
2		leeq1d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		leeq1d.3	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
4		leeq1d.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5	2 1	eqbrtrrd	⊢ ( 𝜑 → 𝐵 ≤ 𝐶 )