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| 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 3922 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
| 2 | 1 | ax-gen 1794 | . . 3 ⊢ ∀𝑦Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | 
| 3 | csbnestgfw 4421 | . . 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 691 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶) | 
| 5 | csbcow 3913 | . . 3 ⊢ ⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶 | |
| 6 | 5 | csbeq2i 3906 | . 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 | 
| 7 | csbcow 3913 | . 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶 | |
| 8 | 4, 6, 7 | 3eqtr3g 2799 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑥⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑥⦌𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 = wceq 1539 ∈ wcel 2107 Ⅎwnfc 2889 ⦋csb 3898 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-v 3481 df-sbc 3788 df-csb 3899 | 
| This theorem is referenced by: csbidm 4432 | 
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