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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 6713, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7198. (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 3886 | . . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹''''𝐵) = ⦋𝐴 / 𝑥⦌(𝐹''''𝐵)) | |
2 | csbeq1 3886 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹) | |
3 | csbeq1 3886 | . . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
4 | 2, 3 | afv2eq12d 43463 | . . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) |
5 | 1, 4 | eqeq12d 2837 | . 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 3497 | . . 3 ⊢ 𝑦 ∈ V | |
7 | nfcsb1v 3907 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹 | |
8 | nfcsb1v 3907 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
9 | 7, 8 | nfafv2 43466 | . . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵) |
10 | csbeq1a 3897 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹) | |
11 | csbeq1a 3897 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
12 | 10, 11 | afv2eq12d 43463 | . . 3 ⊢ (𝑥 = 𝑦 → (𝐹''''𝐵) = (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵)) |
13 | 6, 9, 12 | csbief 3917 | . 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹''''𝐵) = (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵) |
14 | 5, 13 | vtoclg 3567 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⦋csb 3883 ''''cafv2 43456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-dfat 43367 df-afv2 43457 |
This theorem is referenced by: (None) |
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