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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 | dedlema 910 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑)))) | |
2 | 1 | abbi2dv 2197 | . 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))}) |
3 | df-if 3352 | . 2 ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} | |
4 | 2, 3 | syl6reqr 2132 | 1 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∨ wo 661 = wceq 1284 ∈ wcel 1433 {cab 2067 ifcif 3351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-if 3352 |
This theorem is referenced by: iftruei 3357 iftrued 3358 ifsbdc 3363 ifcldadc 3378 ifbothdc 3380 ifcldcd 3381 fidifsnen 6355 uzin 8651 fzprval 9099 fztpval 9100 modifeq2int 9388 expival 9478 bcval 9676 bcval2 9677 flodddiv4 10334 gcd0val 10352 dfgcd2 10403 eucalgf 10437 eucalginv 10438 eucalglt 10439 |
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