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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ififc | Structured version Visualization version GIF version |
Description: A biconditional connecting the conditional operator for propositions and the conditional operator for classes. Note that there is no sethood hypothesis on 𝑋: it is implied by either side. (Contributed by BJ, 24-Sep-2019.) Generalize statement from setvar 𝑥 to class 𝑋. (Revised by BJ, 26-Dec-2023.) |
Ref | Expression |
---|---|
bj-ififc | ⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-df-ifc 34747 | . . 3 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ 𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}) |
3 | df-ifp 1061 | . . . 4 ⊢ (if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑋 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑋 ∈ 𝐵))) | |
4 | elex 3448 | . . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V) | |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ V) |
6 | elex 3448 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ ((¬ 𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ V) |
8 | 5, 7 | jaoi 854 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ V) |
9 | 3, 8 | sylbi 216 | . . 3 ⊢ (if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵) → 𝑋 ∈ V) |
10 | eleq1 2826 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) | |
11 | eleq1 2826 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) | |
12 | 10, 11 | ifpbi23d 1079 | . . 3 ⊢ (𝑥 = 𝑋 → (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵))) |
13 | 9, 12 | elab3 3617 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵)) |
14 | 2, 13 | bitri 274 | 1 ⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∨ wo 844 if-wif 1060 = wceq 1539 ∈ wcel 2106 {cab 2715 Vcvv 3430 ifcif 4460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3432 df-if 4461 |
This theorem is referenced by: (None) |
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