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| Mirrors > Home > MPE Home > Th. List > iuneq2dv | Structured version Visualization version GIF version | ||
| Description: Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.) |
| Ref | Expression |
|---|---|
| iuneq2dv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| iuneq2dv | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iuneq2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 2 | 1 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
| 3 | iuneq2 5011 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) | |
| 4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∪ ciun 4991 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-iun 4993 |
| This theorem is referenced by: iuneq12dOLD 5020 iuneq12d 5021 iuneq2d 5022 fparlem3 8139 fparlem4 8140 oalim 8570 omlim 8571 oelim 8572 oelim2 8633 r1val3 9878 imasdsval 17560 acsfn 17702 tgidm 22987 cmpsub 23408 alexsublem 24052 bcth3 25365 ovoliunlem1 25537 voliunlem1 25585 uniiccdif 25613 uniioombllem2 25618 uniioombllem3a 25619 uniioombllem4 25621 itg2monolem1 25785 taylpfval 26406 dmdju 32657 ofpreima2 32676 fnpreimac 32681 ssdifidllem 33484 esum2dlem 34093 eulerpartlemgu 34379 cvmscld 35278 satom 35361 msubvrs 35565 mblfinlem2 37665 ftc1anclem6 37705 heibor 37828 prjspval2 42623 trclfvcom 43736 scottrankd 44267 meaiininclem 46501 carageniuncllem2 46537 hoidmv1le 46609 hoidmvle 46615 ovnhoilem2 46617 ovnhoi 46618 ovnlecvr2 46625 ovncvr2 46626 hspmbl 46644 ovolval4lem1 46664 ovnovollem1 46671 ovnovollem2 46672 iunhoiioo 46691 vonioolem2 46696 smflimlem4 46789 smflimlem6 46791 |
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