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Mirrors > Home > ILE Home > Th. List > iftrue | GIF version |
Description: Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
iftrue | ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-if 3519 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
2 | dedlema 958 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)))) | |
3 | 2 | abbi2dv 2283 | . 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}) |
4 | 1, 3 | eqtr4id 2216 | 1 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 = wceq 1342 ∈ wcel 2135 {cab 2150 ifcif 3518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-if 3519 |
This theorem is referenced by: iftruei 3524 iftrued 3525 ifsbdc 3530 ifcldadc 3547 ifbothdadc 3549 ifbothdc 3550 ifiddc 3551 ifcldcd 3553 ifandc 3554 fidifsnen 6830 nnnninf 7084 nnnninf2 7085 mkvprop 7116 uzin 9492 fzprval 10011 fztpval 10012 modifeq2int 10315 bcval 10656 bcval2 10657 sumrbdclem 11312 fsum3cvg 11313 summodclem2a 11316 isumss2 11328 fsum3ser 11332 fsumsplit 11342 sumsplitdc 11367 prodrbdclem 11506 fproddccvg 11507 iprodap 11515 iprodap0 11517 prodssdc 11524 fprodsplitdc 11531 flodddiv4 11865 gcd0val 11887 dfgcd2 11941 eucalgf 11981 eucalginv 11982 eucalglt 11983 phisum 12166 pc0 12230 pcgcd 12254 pcmptcl 12266 pcmpt 12267 pcmpt2 12268 pcprod 12270 fldivp1 12272 1arithlem4 12290 unct 12369 dvexp2 13274 nnsf 13778 nninfsellemsuc 13785 |
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