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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 6716, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7201. (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 3889 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = ⦋𝐴 / 𝑥⦌(𝐹'''𝐵)) | |
2 | csbeq1 3889 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹) | |
3 | csbeq1 3889 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
4 | 2, 3 | afveq12d 43339 | . . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵)) |
5 | 1, 4 | eqeq12d 2840 | . 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 3500 | . . 3 ⊢ 𝑦 ∈ V | |
7 | nfcsb1v 3910 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹 | |
8 | nfcsb1v 3910 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
9 | 7, 8 | nfafv 43342 | . . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) |
10 | csbeq1a 3900 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹) | |
11 | csbeq1a 3900 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
12 | 10, 11 | afveq12d 43339 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵)) |
13 | 6, 9, 12 | csbief 3920 | . 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) |
14 | 5, 13 | vtoclg 3570 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⦋csb 3886 '''cafv 43323 |
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-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
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-ne 3020 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-int 4880 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-res 5570 df-iota 6317 df-fun 6360 df-fv 6366 df-aiota 43292 df-dfat 43325 df-afv 43326 |
This theorem is referenced by: (None) |
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