Theorem ditgeq12d 36165

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

Hypotheses

Ref	Expression
ditgeq12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
ditgeq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
ditgeq12d	⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷

Allowed substitution hints:   𝜑(𝑥)   𝐸(𝑥)

Proof of Theorem ditgeq12d

Step	Hyp	Ref	Expression
1		ditgeq12d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ditgeq12d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		ditgeq1 25879	. . 3 ⊢ (𝐴 = 𝐵 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐶]𝐸 d𝑥)
4		ditgeq2 25880	. . 3 ⊢ (𝐶 = 𝐷 → ⨜[𝐵 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
5	3, 4	sylan9eq 2793	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
6	1, 2, 5	syl2anc 583	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)

Syntax hints:   → wi 4   = wceq 1535  ⨜cdit 25877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6317  df-iota 6510  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-neg 11486  df-seq 14029  df-sum 15709  df-itg 25653  df-ditg 25878

This theorem is referenced by: (None)