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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-clelsb3df | Structured version Visualization version GIF version |
Description: Deduction version of clelsb3f 2981. (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 1914 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
2 | clelsb3df.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | clelsb3df.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴) | |
4 | 3 | nfcrd 2968 | . . 3 ⊢ (𝜑 → Ⅎ𝑦 𝑤 ∈ 𝐴) |
5 | 1, 2, 4 | sbco2d 2553 | . 2 ⊢ (𝜑 → ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴)) |
6 | clelsb3 2939 | . . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
7 | 6 | sbbii 2080 | . 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴) |
8 | clelsb3 2939 | . 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) | |
9 | 5, 7, 8 | 3bitr3g 315 | 1 ⊢ (𝜑 → ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1783 [wsb 2068 ∈ wcel 2113 Ⅎwnfc 2960 |
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-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clel 2892 df-nfc 2962 |
This theorem is referenced by: wl-dfrabf 34900 |
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