|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > csbov2g | Structured version Visualization version GIF version | ||
| Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) | 
| Ref | Expression | 
|---|---|
| csbov2g | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | csbov12g 7478 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) | |
| 2 | csbconstg 3917 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | |
| 3 | 2 | oveq1d 7447 | . 2 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) | 
| 4 | 1, 3 | eqtrd 2776 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⦋csb 3898 (class class class)co 7432 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-dm 5694 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: csbnegg 11506 prmgaplem7 17096 matgsum 22444 scmatscm 22520 pm2mpf1lem 22801 pm2mpcoe1 22807 pm2mpmhmlem2 22826 monmat2matmon 22831 divcncf 25483 logbmpt 26832 finxpreclem4 37396 tfsconcatfv 43359 cotrclrcl 43760 ply1mulgsumlem3 48310 ply1mulgsumlem4 48311 ply1mulgsum 48312 | 
| Copyright terms: Public domain | W3C validator |