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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 3480 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
2 | dedlema 954 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)))) | |
3 | 2 | abbi2dv 2259 | . 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}) |
4 | 1, 3 | eqtr4id 2192 | 1 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 = wceq 1332 ∈ wcel 1481 {cab 2126 ifcif 3479 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-if 3480 |
This theorem is referenced by: iftruei 3485 iftrued 3486 ifsbdc 3491 ifcldadc 3506 ifbothdadc 3508 ifbothdc 3509 ifiddc 3510 ifcldcd 3512 ifandc 3513 fidifsnen 6772 nnnninf 7031 mkvprop 7040 uzin 9382 fzprval 9893 fztpval 9894 modifeq2int 10190 bcval 10527 bcval2 10528 sumrbdclem 11178 fsum3cvg 11179 summodclem2a 11182 isumss2 11194 fsum3ser 11198 fsumsplit 11208 sumsplitdc 11233 prodrbdclem 11372 fproddccvg 11373 iprodap 11381 iprodap0 11383 flodddiv4 11667 gcd0val 11685 dfgcd2 11738 eucalgf 11772 eucalginv 11773 eucalglt 11774 unct 11991 dvexp2 12884 nnsf 13374 nninfsellemsuc 13383 |
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