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| Description: Deduction version of nfcsb 3926. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nfcsbd.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfcsbd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| nfcsbd.3 | ⊢ (𝜑 → Ⅎ𝑥𝐵) | 
| Ref | Expression | 
|---|---|
| nfcsbd | ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-csb 3900 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵} | |
| 2 | nfv 1914 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
| 3 | nfcsbd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 4 | nfcsbd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 5 | nfcsbd.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 6 | 5 | nfcrd 2899 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵) | 
| 7 | 3, 4, 6 | nfsbcd 3812 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵) | 
| 8 | 2, 7 | nfabd 2928 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}) | 
| 9 | 1, 8 | nfcxfrd 2904 | 1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 Ⅎwnf 1783 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 [wsbc 3788 ⦋csb 3899 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-sbc 3789 df-csb 3900 | 
| This theorem is referenced by: nfcsb 3926 | 
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