Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfsbcd | Structured version Visualization version GIF version |
Description: Deduction version of nfsbc 3794. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
nfsbcd.1 | ⊢ Ⅎ𝑦𝜑 |
nfsbcd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfsbcd.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfsbcd | ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3770 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
2 | nfsbcd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfsbcd.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfsbcd.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfabd 2998 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
6 | 2, 5 | nfeld 2986 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfxfrd 1845 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1775 ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-sbc 3770 |
This theorem is referenced by: nfsbc 3794 nfcsbd 3905 sbcnestgf 4372 |
Copyright terms: Public domain | W3C validator |