Mathbox for Wolf Lammen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-clelsb3df | Structured version Visualization version GIF version |
Description: Deduction version of clelsb3f 2979. (Contributed by Wolf Lammen, 29-May-2023.) |
Ref | Expression |
---|---|
clelsb3df.1 | ⊢ Ⅎ𝑦𝜑 |
clelsb3df.2 | ⊢ (𝜑 → Ⅎ𝑦𝐴) |
Ref | Expression |
---|---|
wl-clelsb3df | ⊢ (𝜑 → ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
2 | clelsb3df.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | clelsb3df.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴) | |
4 | 3 | nfcrd 2966 | . . 3 ⊢ (𝜑 → Ⅎ𝑦 𝑤 ∈ 𝐴) |
5 | 1, 2, 4 | sbco2d 2547 | . 2 ⊢ (𝜑 → ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴)) |
6 | clelsb3 2937 | . . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
7 | 6 | sbbii 2072 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴) |
8 | clelsb3 2937 | . 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
9 | 5, 7, 8 | 3bitr3g 314 | 1 ⊢ (𝜑 → ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 Ⅎwnf 1775 [wsb 2060 ∈ wcel 2105 Ⅎwnfc 2958 |
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-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
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-clel 2890 df-nfc 2960 |
This theorem is referenced by: wl-dfrabf 34745 |
Copyright terms: Public domain | W3C validator |