Theorem ditgeq12d 36754

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 26007	. . 3 ⊢ (𝐴 = 𝐵 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐶]𝐸 d𝑥)
4		ditgeq2 26008	. . 3 ⊢ (𝐶 = 𝐷 → ⨜[𝐵 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
5	3, 4	sylan9eq 2818	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
6	1, 2, 5	syl2anc 595	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)

Syntax hints:   → wi 4   = wceq 1570  ⨜cdit 26005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-iota 6492  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-neg 11439  df-seq 14034  df-sum 15734  df-itg 25782  df-ditg 26006

This theorem is referenced by: (None)