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Mirrors > Home > MPE Home > Th. List > csbov12g | Structured version Visualization version GIF version |
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) |
Ref | Expression |
---|---|
csbov12g | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbov123 7201 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) | |
2 | csbconstg 3905 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐹 = 𝐹) | |
3 | 2 | oveqd 7176 | . 2 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) |
4 | 1, 3 | syl5eq 2871 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵𝐹𝐶) = (⦋𝐴 / 𝑥⦌𝐵𝐹⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⦋csb 3886 (class class class)co 7159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 ax-pow 5269 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-dm 5568 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: csbov1g 7204 csbov2g 7205 offval22 7786 prmgaplem7 16396 mptscmfsupp0 19702 pm2mp 21436 chfacfscmulfsupp 21470 chfacfpmmulfsupp 21474 cayhamlem4 21499 iccelpart 43600 ply1mulgsumlem4 44450 |
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