Theorem ditgeq3sdv 36588

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 2765	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
2		eqidd 2765	. 2 ⊢ (𝜑 → 𝐵 = 𝐵)
3		ditgeq3sdv.1	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
4	1, 2, 3	ditgeq123dv 36586	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥)

Syntax hints:   → wi 4   = wceq 1562  ⨜cdit 25910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5655  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-iota 6479  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-neg 11419  df-seq 14017  df-sum 15716  df-itg 25687  df-ditg 25911

This theorem is referenced by: (None)