Theorem ditgeq12d 36169

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 25788	. . 3 ⊢ (𝐴 = 𝐵 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐶]𝐸 d𝑥)
4		ditgeq2 25789	. . 3 ⊢ (𝐶 = 𝐷 → ⨜[𝐵 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
5	3, 4	sylan9eq 2789	. 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 1539  ⨜cdit 25786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-xp 5658  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6288  df-iota 6481  df-fv 6536  df-ov 7403  df-oprab 7404  df-mpo 7405  df-frecs 8275  df-wrecs 8306  df-recs 8380  df-rdg 8419  df-neg 11462  df-seq 14010  df-sum 15692  df-itg 25563  df-ditg 25787

This theorem is referenced by: (None)