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Mirrors > Home > MPE Home > Th. List > dfif2 | Structured version Visualization version GIF version |
Description: An alternate definition of the conditional operator df-if 4037 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.) |
Ref | Expression |
---|---|
dfif2 | ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-if 4037 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
2 | df-or 384 | . . . 4 ⊢ (((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
3 | orcom 401 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐵 ∧ ¬ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
4 | iman 439 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) | |
5 | 4 | imbi1i 338 | . . . 4 ⊢ (((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (¬ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))) |
6 | 2, 3, 5 | 3bitr4i 291 | . . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))) |
7 | 6 | abbii 2726 | . 2 ⊢ {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))} |
8 | 1, 7 | eqtri 2632 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐵 → 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜑))} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ifcif 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-if 4037 |
This theorem is referenced by: iftrue 4042 nfifd 4064 |
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