Theorem ditgeq3dv

Description: Equality theorem for the directed integral. (Contributed by Mario Carneiro, 13-Aug-2014)

	Ref	Expression
Hypothesis	ditgeq3dv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝐷 = 𝐸 )
Assertion	ditgeq3dv	⊢ ( 𝜑 → ⨜ [ 𝐴 → 𝐵 ] 𝐷 d 𝑥 = ⨜ [ 𝐴 → 𝐵 ] 𝐸 d 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ditgeq3dv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝐷 = 𝐸 )
2	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ 𝐷 = 𝐸 )
3		ditgeq3	⊢ ( ∀ 𝑥 ∈ ℝ 𝐷 = 𝐸 → ⨜ [ 𝐴 → 𝐵 ] 𝐷 d 𝑥 = ⨜ [ 𝐴 → 𝐵 ] 𝐸 d 𝑥 )
4	2 3	syl	⊢ ( 𝜑 → ⨜ [ 𝐴 → 𝐵 ] 𝐷 d 𝑥 = ⨜ [ 𝐴 → 𝐵 ] 𝐸 d 𝑥 )