Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > csbafv212g | Structured version Visualization version GIF version |
Description: Move class substitution in and out of a function value, analogous to csbfv12 6749, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7244. (Contributed by AV, 4-Sep-2022.) |
Ref | Expression |
---|---|
csbafv212g | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3805 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹''''𝐵) = ⦋𝐴 / 𝑥⦌(𝐹''''𝐵)) | |
2 | csbeq1 3805 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹) | |
3 | csbeq1 3805 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
4 | 2, 3 | afv2eq12d 44333 | . . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) |
5 | 1, 4 | eqeq12d 2750 | . 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 3405 | . . 3 ⊢ 𝑦 ∈ V | |
7 | nfcsb1v 3827 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹 | |
8 | nfcsb1v 3827 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
9 | 7, 8 | nfafv2 44336 | . . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵) |
10 | csbeq1a 3816 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹) | |
11 | csbeq1a 3816 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
12 | 10, 11 | afv2eq12d 44333 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐹''''𝐵) = (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵)) |
13 | 6, 9, 12 | csbief 3837 | . 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹''''𝐵) = (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵) |
14 | 5, 13 | vtoclg 3474 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ⦋csb 3802 ''''cafv2 44326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-iota 6327 df-fun 6371 df-dfat 44237 df-afv2 44327 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |