Theorem ditgeq3sdv 37012

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 2761	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
2		eqidd 2761	. 2 ⊢ (𝜑 → 𝐵 = 𝐵)
3		ditgeq3sdv.1	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
4	1, 2, 3	ditgeq123dv 37010	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ⨜cdit 26166

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-xp 5656  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-iota 6494  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-neg 11544  df-seq 14145  df-sum 15854  df-itg 25944  df-ditg 26167

This theorem is used by: (None)