Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfsbcdw | Structured version Visualization version GIF version |
Description: Deduction version of nfsbcw 3790. Version of nfsbcd 3792 with a disjoint variable condition, which does not require ax-13 2389. (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 3769 | . 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 2999 | . . 3 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
6 | 2, 5 | nfeld 2988 | . 2 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ {𝑦 ∣ 𝜓}) |
7 | 1, 6 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1783 ∈ wcel 2113 {cab 2798 Ⅎwnfc 2960 [wsbc 3768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-sbc 3769 |
This theorem is referenced by: nfsbcw 3790 nfcsbw 3902 sbcnestgfw 4363 |
Copyright terms: Public domain | W3C validator |