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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 2778 from wl-clabv 34379. (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 2866 | . 2 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑})) | |
2 | wl-clabv 34379 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
3 | sbequ 2065 | . . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑦]𝜑 ↔ [𝑥 / 𝑦]𝜑)) | |
4 | 2, 3 | syl5bb 284 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)) |
5 | 4 | pm5.32i 575 | . . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) |
6 | 5 | exbii 1833 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) |
7 | 19.41v 1931 | . . 3 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) | |
8 | ax6ev 1953 | . . . 4 ⊢ ∃𝑧 𝑧 = 𝑥 | |
9 | 8 | biantrur 531 | . . 3 ⊢ ([𝑥 / 𝑦]𝜑 ↔ (∃𝑧 𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑)) |
10 | 7, 9 | bitr4i 279 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ [𝑥 / 𝑦]𝜑) ↔ [𝑥 / 𝑦]𝜑) |
11 | 1, 6, 10 | 3bitri 298 | 1 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1765 [wsb 2044 ∈ wcel 2083 {cab 2777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1766 df-sb 2045 df-clab 2778 df-clel 2865 |
This theorem is referenced by: (None) |
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