Theorem ditgeq3sdv 36181

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

Hypothesis

Ref	Expression
ditgeq3sdv.1	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
ditgeq3sdv	⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem ditgeq3sdv

Step	Hyp	Ref	Expression
1		eqidd 2741	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
2		eqidd 2741	. 2 ⊢ (𝜑 → 𝐵 = 𝐵)
3		ditgeq3sdv.1	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
4	1, 2, 3	ditgeq123dv 36179	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥)

Syntax hints:   → wi 4   = wceq 1537  ⨜cdit 25893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5701  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-pred 6327  df-iota 6520  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-frecs 8316  df-wrecs 8347  df-recs 8421  df-rdg 8460  df-neg 11517  df-seq 14047  df-sum 15729  df-itg 25669  df-ditg 25894

This theorem is referenced by: (None)