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Mirrors > Home > MPE Home > Th. List > nfsbc1d | Structured version Visualization version GIF version |
Description: Deduction version of nfsbc1 3794. (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 3776 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
2 | nfsbc1d.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfab1 2894 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑥 ∣ 𝜓}) |
5 | 2, 4 | nfeld 2904 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜓}) |
6 | 1, 5 | nfxfrd 1849 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1778 ∈ wcel 2099 {cab 2703 Ⅎwnfc 2876 [wsbc 3775 |
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 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-sbc 3776 |
This theorem is referenced by: nfsbc1 3794 nfcsb1d 3914 |
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