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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-dfclab | Structured version Visualization version GIF version |
Description: Rederive df-clab 2702 from wl-clabv 36913. (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 2803 | . 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑})) | |
2 | wl-clabv 36913 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
3 | sbequ 2078 | . . . . 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 1842 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) |
7 | 19.41v 1945 | . . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) | |
8 | ax6ev 1965 | . . . 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 205 ∧ wa 395 ∃wex 1773 [wsb 2059 ∈ wcel 2098 {cab 2701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 df-clab 2702 df-clel 2802 |
This theorem is referenced by: (None) |
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