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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-clabt | Structured version Visualization version GIF version |
Description: Using class abstraction in a context. For a version based on fewer axioms see wl-clabtv 34994. (Contributed by Wolf Lammen, 29-May-2023.) |
Ref | Expression |
---|---|
wl-clabt.nf | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
wl-clabt | ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-clabt.nf | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | biimt 364 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓))) | |
3 | 1, 2 | sbbid 2244 | . . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥](𝜑 → 𝜓))) |
4 | df-clab 2777 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
5 | df-clab 2777 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)} ↔ [𝑦 / 𝑥](𝜑 → 𝜓)) | |
6 | 3, 4, 5 | 3bitr4g 317 | . 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 → 𝜓)})) |
7 | 6 | eqrdv 2796 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ (𝜑 → 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 Ⅎwnf 1785 [wsb 2069 ∈ wcel 2111 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 |
This theorem is referenced by: (None) |
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