Theorem int-ineqtransd

Description: InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-ineqtransd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	int-ineqtransd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	int-ineqtransd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	int-ineqtransd.4	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
	int-ineqtransd.5	⊢ ( 𝜑 → 𝐶 ≤ 𝐵 )
Assertion	int-ineqtransd	⊢ ( 𝜑 → 𝐶 ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		int-ineqtransd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		int-ineqtransd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		int-ineqtransd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		int-ineqtransd.4	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
5		int-ineqtransd.5	⊢ ( 𝜑 → 𝐶 ≤ 𝐵 )
6	3 2 1 5 4	letrd	⊢ ( 𝜑 → 𝐶 ≤ 𝐴 )