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Mirrors > Home > MPE Home > Th. List > cbviunv | Structured version Visualization version GIF version |
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.) Add disjoint variable condition to avoid ax-13 2386. See cbviunvg 4959 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
Ref | Expression |
---|---|
cbviunv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbviunv | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2977 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2977 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbviunv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbviun 4953 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∪ ciun 4911 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-iun 4913 |
This theorem is referenced by: iunxdif2 4969 otiunsndisj 5402 onfununi 7972 oelim2 8215 marypha2lem2 8894 trcl 9164 r1om 9660 fictb 9661 cfsmolem 9686 cfsmo 9687 domtriomlem 9858 domtriom 9859 pwfseq 10080 wunex2 10154 wuncval2 10163 fsuppmapnn0fiubex 13354 s3iunsndisj 14322 ackbijnn 15177 smndex1basss 18064 smndex1mgm 18066 efgs1b 18856 ablfaclem3 19203 ptbasfi 22183 bcth3 23928 itg1climres 24309 suppovss 30420 hashunif 30522 bnj601 32187 cvmliftlem15 32540 trpredlem1 33061 trpredtr 33064 trpredmintr 33065 trpredrec 33072 neibastop2 33704 filnetlem4 33724 sstotbnd2 35046 heiborlem3 35085 heibor 35093 lcfr 38715 mapdrval 38777 corclrcl 40045 trclrelexplem 40049 dftrcl3 40058 cotrcltrcl 40063 dfrtrcl3 40071 corcltrcl 40077 cotrclrcl 40080 ssmapsn 41472 cnrefiisplem 42103 cnrefiisp 42104 meaiuninclem 42756 meaiuninc 42757 meaiininc 42763 carageniuncllem2 42798 caratheodorylem1 42802 caratheodorylem2 42803 caratheodory 42804 ovnsubadd 42848 hoidmv1le 42870 hoidmvle 42876 ovnhoilem2 42878 hspmbl 42905 ovnovollem3 42934 vonvolmbl 42937 smflimlem2 43042 smflimlem3 43043 smflimlem4 43044 smflim 43047 smflim2 43074 smflimsup 43096 otiunsndisjX 43472 |
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