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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbafv12g | Structured version Visualization version GIF version |
Description: Move class substitution in and out of a function value, analogous to csbfv12 6688, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7177. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
Ref | Expression |
---|---|
csbafv12g | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3831 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = ⦋𝐴 / 𝑥⦌(𝐹'''𝐵)) | |
2 | csbeq1 3831 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹) | |
3 | csbeq1 3831 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
4 | 2, 3 | afveq12d 43689 | . . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵)) |
5 | 1, 4 | eqeq12d 2814 | . 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) ↔ ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))) |
6 | vex 3444 | . . 3 ⊢ 𝑦 ∈ V | |
7 | nfcsb1v 3852 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹 | |
8 | nfcsb1v 3852 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
9 | 7, 8 | nfafv 43692 | . . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) |
10 | csbeq1a 3842 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹) | |
11 | csbeq1a 3842 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
12 | 10, 11 | afveq12d 43689 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵)) |
13 | 6, 9, 12 | csbief 3862 | . 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) |
14 | 5, 13 | vtoclg 3515 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⦋csb 3828 '''cafv 43673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-aiota 43642 df-dfat 43675 df-afv 43676 |
This theorem is referenced by: (None) |
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