Theorem ditgeq3sdv 36425

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 2738	. 2 ⊢ (𝜑 → 𝐴 = 𝐴)
2		eqidd 2738	. 2 ⊢ (𝜑 → 𝐵 = 𝐵)
3		ditgeq3sdv.1	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
4	1, 2, 3	ditgeq123dv 36423	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥)

Syntax hints:   → wi 4   = wceq 1542  ⨜cdit 25827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5632  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-iota 6450  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-neg 11375  df-seq 13959  df-sum 15644  df-itg 25604  df-ditg 25828

This theorem is referenced by: (None)