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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 3493 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | df-csb 3895 | . . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} | |
3 | sbcel2gv 3850 | . . . . . . . 8 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | |
4 | 3 | eqabcdv 2869 | . . . . . . 7 ⊢ (𝑦 ∈ V → {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝑥} = 𝑦) |
5 | 2, 4 | eqtrid 2785 | . . . . . 6 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝑥 = 𝑦) |
6 | 5 | elv 3481 | . . . . 5 ⊢ ⦋𝑦 / 𝑥⦌𝑥 = 𝑦 |
7 | 6 | csbeq2i 3902 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑦⦌𝑦 |
8 | csbcow 3909 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝑥 = ⦋𝐴 / 𝑥⦌𝑥 | |
9 | df-csb 3895 | . . . 4 ⊢ ⦋𝐴 / 𝑦⦌𝑦 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} | |
10 | 7, 8, 9 | 3eqtr3i 2769 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝑥 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} |
11 | sbcel2gv 3850 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴)) | |
12 | 11 | eqabcdv 2869 | . . 3 ⊢ (𝐴 ∈ V → {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝑦} = 𝐴) |
13 | 10, 12 | eqtrid 2785 | . 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 1542 ∈ wcel 2107 {cab 2710 Vcvv 3475 [wsbc 3778 ⦋csb 3894 |
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-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-sbc 3779 df-csb 3895 |
This theorem is referenced by: csbvargi 4433 sbccsb2 4435 2nreu 4442 csbfv 6942 ixpsnval 8894 csbwrdg 14494 swrdspsleq 14615 prmgaplem7 16990 telgsums 19861 ixpsnbasval 20832 scmatscm 22015 pm2mpf1lem 22296 pm2mpcoe1 22302 idpm2idmp 22303 pm2mpmhmlem2 22321 monmat2matmon 22326 pm2mp 22327 fvmptnn04if 22351 chfacfscmulfsupp 22361 cayhamlem4 22390 divcncf 24964 opsbc2ie 31716 esum2dlem 33090 relowlpssretop 36245 rdgeqoa 36251 renegclALT 37833 cdlemk40 39788 tfsconcatfv 42091 iscard4 42284 minregex 42285 cotrclrcl 42493 frege124d 42512 frege70 42684 frege72 42686 frege77 42691 frege91 42705 frege92 42706 frege116 42730 frege118 42732 frege120 42734 rusbcALT 43198 onfrALTlem5 43303 onfrALTlem4 43304 onfrALTlem5VD 43646 iccelpart 46101 ply1mulgsumlem4 47070 |
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