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Theorem itgeq1d 45913
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 25823 . 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 1537  citg 25667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seq 14040  df-sum 15720  df-itg 25672
This theorem is referenced by:  itgspltprt  45935  fourierdlem73  46135  fourierdlem81  46143  fourierdlem92  46154  fourierdlem93  46155  fourierdlem103  46165  fourierdlem104  46166  fourierdlem107  46169  fourierdlem109  46171  fourierdlem111  46173
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