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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dfclab | Structured version Visualization version GIF version | ||
| Description: Rederive df-clab 2715 from wl-clabv 37596. (Contributed by Wolf Lammen, 31-May-2023.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| wl-dfclab | ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfclel 2817 | . 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑})) | |
| 2 | wl-clabv 37596 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
| 3 | sbequ 2083 | . . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑)) | |
| 4 | 2, 3 | bitrid 283 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)) |
| 5 | 4 | pm5.32i 574 | . . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) |
| 6 | 5 | exbii 1848 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) |
| 7 | 19.41v 1949 | . . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) | |
| 8 | ax6ev 1969 | . . . 4 ⊢ ∃𝑧 𝑧 = 𝑥 | |
| 9 | 8 | biantrur 530 | . . 3 ⊢ ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) |
| 10 | 7, 9 | bitr4i 278 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ [𝑥 / 𝑦]𝜑) |
| 11 | 1, 6, 10 | 3bitri 297 | 1 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1779 [wsb 2064 ∈ wcel 2108 {cab 2714 |
| 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 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-clel 2816 |
| This theorem is referenced by: (None) |
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