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Mirrors > Home > MPE Home > Th. List > nfsbc1d | Structured version Visualization version GIF version |
Description: Deduction version of nfsbc1 3652. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
nfsbc1d.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Ref | Expression |
---|---|
nfsbc1d | ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3634 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
2 | nfsbc1d.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfab1 2943 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑥 ∣ 𝜓}) |
5 | 2, 4 | nfeld 2950 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜓}) |
6 | 1, 5 | nfxfrd 1950 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1879 ∈ wcel 2157 {cab 2785 Ⅎwnfc 2928 [wsbc 3633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-sbc 3634 |
This theorem is referenced by: nfsbc1 3652 nfcsb1d 3742 |
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