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Mirrors > Home > MPE Home > Th. List > nfsbcdw | Structured version Visualization version GIF version |
Description: Deduction version of nfsbcw 3704. Version of nfsbcd 3706 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfsbcdw.1 | ⊢ Ⅎ𝑦𝜑 |
nfsbcdw.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfsbcdw.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfsbcdw | ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3683 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
2 | nfsbcdw.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfsbcdw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
4 | nfsbcdw.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfabdw 2922 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
6 | 2, 5 | nfeld 2910 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfxfrd 1860 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1790 ∈ wcel 2113 {cab 2716 Ⅎwnfc 2879 [wsbc 3682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-sbc 3683 |
This theorem is referenced by: nfsbcw 3704 nfcsbw 3817 sbcnestgfw 4309 |
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