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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 5271 | . 2 ⊢ (∀𝑥 𝐵 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 ∈ V) | |
2 | csbex.1 | . 2 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | mpg 1800 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3447 ⦋csb 3859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-nul 4287 |
This theorem is referenced by: iunopeqop 5482 dfmpo 8038 cantnfdm 9608 cantnff 9618 bpolylem 15939 ruclem1 16121 pcmpt 16772 cidffn 17566 issubc 17729 natffn 17844 fnxpc 18072 evlfcl 18119 odf 19327 selvval 21551 itgfsum 25214 itgparts 25434 vmaf 26491 mulsval 27403 ttgval 27866 ttgvalOLD 27867 abfmpel 31624 msrf 34200 rdgssun 35899 finxpreclem2 35911 poimirlem17 36145 poimirlem23 36151 poimirlem24 36152 unirep 36222 cdlemk40 39430 aomclem6 41433 rnghmfn 46278 rngchomrnghmresALTV 46384 |
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