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Mirrors > Home > ILE Home > Th. List > iffalse | GIF version |
Description: Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.) |
Ref | Expression |
---|---|
iffalse | ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedlemb 877 | . . 3 ⊢ (¬ 𝜑 → (𝑥 ∈ 𝐵 ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)))) | |
2 | 1 | abbi2dv 2156 | . 2 ⊢ (¬ 𝜑 → 𝐵 = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}) |
3 | df-if 3332 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
4 | 2, 3 | syl6reqr 2091 | 1 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∨ wo 629 = wceq 1243 ∈ wcel 1393 {cab 2026 ifcif 3331 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-if 3332 |
This theorem is referenced by: iffalsei 3340 iffalsed 3341 ifnefalse 3342 ifbothdc 3357 ifcldcd 3358 fidifsnen 6331 uzin 8505 expival 9257 |
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