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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 922 | . . 3 ⊢ (¬ 𝜑 → (𝑥 ∈ 𝐵 ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)))) | |
2 | 1 | abbi2dv 2218 | . 2 ⊢ (¬ 𝜑 → 𝐵 = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}) |
3 | df-if 3422 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
4 | 2, 3 | syl6reqr 2151 | 1 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 670 = wceq 1299 ∈ wcel 1448 {cab 2086 ifcif 3421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-if 3422 |
This theorem is referenced by: iffalsei 3430 iffalsed 3431 ifnefalse 3432 ifsbdc 3433 ifcldadc 3448 ifeq1dadc 3449 ifbothdadc 3450 ifbothdc 3451 ifiddc 3452 ifcldcd 3454 ifandc 3455 fidifsnen 6693 nnnninf 6935 uzin 9208 modifeq2int 10000 bcval 10336 bcval3 10338 sumrbdclem 10984 fsum3cvg 10985 summodclem2a 10989 sumsplitdc 11040 flodddiv4 11426 gcdn0val 11445 dfgcd2 11495 lcmn0val 11540 |
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