Theorem itgeq1d 45944

Description: Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
itgeq1d.aeqb	⊢ (𝜑 → 𝐴 = 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

Step	Hyp	Ref	Expression
1		itgeq1d.aeqb	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		itgeq1 25745	. 2 ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)

Syntax hints:   → wi 4   = wceq 1539  ∫citg 25590

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 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-xp 5671  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-iota 6494  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-seq 14025  df-sum 15706  df-itg 25595

This theorem is referenced by:  itgspltprt  45966  fourierdlem73  46166  fourierdlem81  46174  fourierdlem92  46185  fourierdlem93  46186  fourierdlem103  46196  fourierdlem104  46197  fourierdlem107  46200  fourierdlem109  46202  fourierdlem111  46204