Theorem ditgeq12d 36203

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 25782	. . 3 ⊢ (𝐴 = 𝐵 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐶]𝐸 d𝑥)
4		ditgeq2 25783	. . 3 ⊢ (𝐶 = 𝐷 → ⨜[𝐵 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
5	3, 4	sylan9eq 2784	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
6	1, 2, 5	syl2anc 584	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)

Syntax hints:   → wi 4   = wceq 1540  ⨜cdit 25780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-iota 6452  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-neg 11384  df-seq 13943  df-sum 15629  df-itg 25557  df-ditg 25781

This theorem is referenced by: (None)