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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 33810 | . . 3 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} | |
2 | 1 | eleq2i 2901 | . 2 ⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ 𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)}) |
3 | df-ifp 1055 | . . . 4 ⊢ (if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑋 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑋 ∈ 𝐵))) | |
4 | elex 3510 | . . . . . 6 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V) | |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ V) |
6 | elex 3510 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ ((¬ 𝜑 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ V) |
8 | 5, 7 | jaoi 851 | . . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑋 ∈ 𝐵)) → 𝑋 ∈ V) |
9 | 3, 8 | sylbi 218 | . . 3 ⊢ (if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵) → 𝑋 ∈ V) |
10 | biidd 263 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜑)) | |
11 | eleq1 2897 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) | |
12 | eleq1 2897 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) | |
13 | 10, 11, 12 | ifpbi123d 1069 | . . 3 ⊢ (𝑥 = 𝑋 → (if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵))) |
14 | 9, 13 | elab3 3671 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ if-(𝜑, 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵)} ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵)) |
15 | 2, 14 | bitri 276 | 1 ⊢ (𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∨ wo 841 if-wif 1054 = wceq 1528 ∈ wcel 2105 {cab 2796 Vcvv 3492 ifcif 4463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-if 4464 |
This theorem is referenced by: (None) |
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