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Mirrors > Home > MPE Home > Th. List > csbvarg | Structured version Visualization version GIF version |
Description: The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
csbvarg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3450 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | df-csb 3807 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} | |
3 | sbcel2gv 3764 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
4 | 3 | abbi1dv 2919 | . . . . . . 7 ⊢ (𝑦 ∈ V → {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} = 𝑦) |
5 | 2, 4 | syl5eq 2841 | . . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦) |
6 | 5 | elv 3437 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 |
7 | 6 | csbeq2i 3814 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑦⦌𝑦 |
8 | csbco 3820 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑥⦌𝑥 | |
9 | df-csb 3807 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌𝑦 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} | |
10 | 7, 8, 9 | 3eqtr3i 2825 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝑥 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} |
11 | sbcel2gv 3764 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴)) | |
12 | 11 | abbi1dv 2919 | . . 3 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} = 𝐴) |
13 | 10, 12 | syl5eq 2841 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) |
14 | 1, 13 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 ∈ wcel 2079 {cab 2773 Vcvv 3432 [wsbc 3701 ⦋csb 3806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-v 3434 df-sbc 3702 df-csb 3807 |
This theorem is referenced by: csbvargi 4293 sbccsb2 4295 2nreu 4301 csbfv 6575 ixpsnval 8303 csbwrdg 13729 swrdspsleq 13851 prmgaplem7 16210 telgsums 18818 ixpsnbasval 19659 scmatscm 20794 pm2mpf1lem 21074 pm2mpcoe1 21080 idpm2idmp 21081 pm2mpmhmlem2 21099 monmat2matmon 21104 pm2mp 21105 fvmptnn04if 21129 chfacfscmulfsupp 21139 cayhamlem4 21168 divcncf 23719 opsbc2ie 29918 esum2dlem 30924 relowlpssretop 34122 rdgeqoa 34128 renegclALT 35580 cdlemk40 37534 iscard4 39336 cotrclrcl 39523 frege124d 39542 frege70 39715 frege72 39717 frege77 39722 frege91 39736 frege92 39737 frege116 39761 frege118 39763 frege120 39765 rusbcALT 40261 onfrALTlem5 40367 onfrALTlem4 40368 onfrALTlem5VD 40710 iccelpart 43029 ply1mulgsumlem4 43877 |
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