Theorem ditgeq12d 37188

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 26195	. . 3 ⊢ (𝐴 = 𝐵 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐶]𝐸 d𝑥)
4		ditgeq2 26196	. . 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 596	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⨜cdit 26193

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-xp 5661  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-iota 6509  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-neg 11568  df-seq 14169  df-sum 15880  df-itg 25971  df-ditg 26194

This theorem is used by: (None)