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| Description: Equality theorem for an integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.) | 
| Ref | Expression | 
|---|---|
| itgeq1d.aeqb | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| itgeq1d | ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | itgeq1d.aeqb | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | itgeq1 25809 | . 2 ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥) | |
| 3 | 1, 2 | syl 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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