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Mirrors > Home > MPE Home > Th. List > cbvitgv | Structured version Visualization version GIF version |
Description: Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
cbvitg.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvitgv | ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvitg.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
2 | nfcv 2902 | . 2 ⊢ Ⅎ𝑦𝐵 | |
3 | nfcv 2902 | . 2 ⊢ Ⅎ𝑥𝐶 | |
4 | 1, 2, 3 | cbvitg 23741 | 1 ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∫citg 23586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-seq 12996 df-sum 14616 df-itg 23591 |
This theorem is referenced by: ftc1a 23999 tgoldbachgtd 31049 itgiccshift 40699 itgperiod 40700 dirkeritg 40822 fourierdlem73 40899 fourierdlem82 40908 fourierdlem93 40919 fourierdlem111 40937 fourierdlem112 40938 |
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