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Mirrors > Home > MPE Home > Th. List > nfcsbd | Structured version Visualization version GIF version |
Description: Deduction version of nfcsb 3917. Usage of this theorem is discouraged because it depends on ax-13 2366. (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 3890 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵} | |
2 | nfv 1910 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfcsbd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfcsbd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfcsbd.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | 5 | nfcrd 2887 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵) |
7 | 3, 4, 6 | nfsbcd 3798 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵) |
8 | 2, 7 | nfabd 2923 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}) |
9 | 1, 8 | nfcxfrd 2897 | 1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1778 ∈ wcel 2099 {cab 2704 Ⅎwnfc 2878 [wsbc 3774 ⦋csb 3889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-13 2366 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-sbc 3775 df-csb 3890 |
This theorem is referenced by: nfcsb 3917 |
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