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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 2800. We reprove the current df-if 4467 from it in bj-dfif 33909. (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 4467 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
2 | ancom 463 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝜑 ∧ 𝑥 ∈ 𝐴)) | |
3 | ancom 463 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ↔ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵)) | |
4 | 2, 3 | orbi12i 911 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))) |
5 | df-ifp 1058 | . . . 4 ⊢ (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))) | |
6 | 4, 5 | bitr4i 280 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)) |
7 | 6 | abbii 2886 | . 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} |
8 | 1, 7 | eqtri 2844 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∨ wo 843 if-wif 1057 = wceq 1533 ∈ wcel 2110 {cab 2799 ifcif 4466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-if 4467 |
This theorem is referenced by: bj-dfif 33909 bj-ififc 33910 |
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