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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-df-ifc | Structured version Visualization version GIF version | ||
| Description: Candidate definition for the conditional operator for classes. This is in line with the definition of a class as the extension of a predicate in df-clab 2715. We reprove the current df-if 4526 from it in bj-dfif 36582. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-df-ifc | ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-if 4526 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
| 2 | ancom 460 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴)) | |
| 3 | ancom 460 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ↔ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)) | |
| 4 | 2, 3 | orbi12i 915 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))) |
| 5 | df-ifp 1064 | . . . 4 ⊢ (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))) | |
| 6 | 4, 5 | bitr4i 278 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)) |
| 7 | 6 | abbii 2809 | . 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} |
| 8 | 1, 7 | eqtri 2765 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 848 if-wif 1063 = wceq 1540 ∈ wcel 2108 {cab 2714 ifcif 4525 |
| 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-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-if 4526 |
| This theorem is referenced by: bj-dfif 36582 bj-ififc 36583 |
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