Theorem leeq2d

Description: Specialization of breq2d to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	leeq2d.1	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
	leeq2d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
	leeq2d.3	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	leeq2d.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion	leeq2d	⊢ ( 𝜑 → 𝐴 ≤ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		leeq2d.1	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
2		leeq2d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		leeq2d.3	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
4		leeq2d.4	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
5	1 2	breqtrd	⊢ ( 𝜑 → 𝐴 ≤ 𝐷 )