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Mirrors > Home > MPE Home > Th. List > csbex | Structured version Visualization version GIF version |
Description: The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Revised by NM, 17-Aug-2018.) |
Ref | Expression |
---|---|
csbex.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
csbex | ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbexg 4720 | . 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | |
2 | csbex.1 | . 2 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | mpg 1715 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ⦋csb 3499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-nul 3875 |
This theorem is referenced by: iunopeqop 4906 dfmpt2 7154 cantnfdm 8444 cantnff 8454 bpolylem 14618 ruclem1 14799 pcmpt 15434 cidffn 16162 issubc 16318 natffn 16432 fnxpc 16639 evlfcl 16685 odf 17779 itgfsum 23399 itgparts 23614 vmaf 24645 ttgval 25555 abfmpel 28835 msrf 30693 finxpreclem2 32403 poimirlem17 32596 poimirlem23 32602 poimirlem24 32603 unirep 32677 cdlemk40 35223 aomclem6 36647 rnghmfn 41680 rngchomrnghmresALTV 41788 |
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