Theorem ditgeq12d

Description: Equality theorem for the directed integral. Deduction form. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	ditgeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	ditgeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	ditgeq12d	⊢ ( 𝜑 → ⨜ [ 𝐴 → 𝐶 ] 𝐸 d 𝑥 = ⨜ [ 𝐵 → 𝐷 ] 𝐸 d 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ditgeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		ditgeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		ditgeq1	⊢ ( 𝐴 = 𝐵 → ⨜ [ 𝐴 → 𝐶 ] 𝐸 d 𝑥 = ⨜ [ 𝐵 → 𝐶 ] 𝐸 d 𝑥 )
4		ditgeq2	⊢ ( 𝐶 = 𝐷 → ⨜ [ 𝐵 → 𝐶 ] 𝐸 d 𝑥 = ⨜ [ 𝐵 → 𝐷 ] 𝐸 d 𝑥 )
5	3 4	sylan9eq	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ⨜ [ 𝐴 → 𝐶 ] 𝐸 d 𝑥 = ⨜ [ 𝐵 → 𝐷 ] 𝐸 d 𝑥 )
6	1 2 5	syl2anc	⊢ ( 𝜑 → ⨜ [ 𝐴 → 𝐶 ] 𝐸 d 𝑥 = ⨜ [ 𝐵 → 𝐷 ] 𝐸 d 𝑥 )