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Mirrors > Home > MPE Home > Th. List > csbnest1g | Structured version Visualization version GIF version |
Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) |
Ref | Expression |
---|---|
csbnest1g | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcsb1v 3745 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
2 | 1 | ax-gen 1891 | . . 3 ⊢ ∀𝑦Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 |
3 | csbnestgf 4192 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶) | |
4 | 2, 3 | mpan2 683 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶) |
5 | csbco 3739 | . . 3 ⊢ ⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶 | |
6 | 5 | csbeq2i 4189 | . 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 |
7 | csbco 3739 | . 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶 | |
8 | 4, 6, 7 | 3eqtr3g 2857 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1651 = wceq 1653 ∈ wcel 2157 Ⅎwnfc 2929 ⦋csb 3729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-v 3388 df-sbc 3635 df-csb 3730 |
This theorem is referenced by: csbidm 4198 |
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