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Mirrors > Home > MPE Home > Th. List > nfcsbd | Structured version Visualization version GIF version |
Description: Deduction version of nfcsb 3909. Usage of this theorem is discouraged because it depends on ax-13 2386. (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 3883 | . 2 ⊢ ⦋𝐴 / 𝑦⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵} | |
2 | nfv 1911 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
3 | nfcsbd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfcsbd.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfcsbd.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | 5 | nfcrd 2969 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐵) |
7 | 3, 4, 6 | nfsbcd 3795 | . . 3 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝑧 ∈ 𝐵) |
8 | 2, 7 | nfabd 3001 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ 𝐵}) |
9 | 1, 8 | nfcxfrd 2976 | 1 ⊢ (𝜑 → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1780 ∈ wcel 2110 {cab 2799 Ⅎwnfc 2961 [wsbc 3771 ⦋csb 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sbc 3772 df-csb 3883 |
This theorem is referenced by: nfcsb 3909 |
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