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| Description: Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 2376. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| sbcco3g.1 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| csbco3g | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | csbnestg 4428 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐷) | |
| 2 | elex 3500 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 3 | nfcvd 2905 | . . . . 5 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶) | |
| 4 | sbcco3g.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
| 5 | 3, 4 | csbiegf 3931 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) | 
| 6 | 2, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) | 
| 7 | 6 | csbeq1d 3902 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) | 
| 8 | 1, 7 | eqtrd 2776 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⦋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-13 2376 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: (None) | 
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