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Theorem itgeq1d 45977
Description: Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
itgeq1d.aeqb (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
itgeq1d (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem itgeq1d
StepHypRef Expression
1 itgeq1d.aeqb . 2 (𝜑𝐴 = 𝐵)
2 itgeq1 25809 . 2 (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
31, 2syl 17 1 (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  citg 25654
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 2707
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 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-xp 5690  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seq 14044  df-sum 15724  df-itg 25659
This theorem is referenced by:  itgspltprt  45999  fourierdlem73  46199  fourierdlem81  46207  fourierdlem92  46218  fourierdlem93  46219  fourierdlem103  46229  fourierdlem104  46230  fourierdlem107  46233  fourierdlem109  46235  fourierdlem111  46237
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