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Theorem iftrue 3336
 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.)
Assertion
Ref Expression
iftrue (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)

Proof of Theorem iftrue
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dedlema 876 . . 3 (𝜑 → (𝑥𝐴 ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))))
21abbi2dv 2156 . 2 (𝜑𝐴 = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))})
3 df-if 3332 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
42, 3syl6reqr 2091 1 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
 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:  iftruei  3337  iftrued  3338  ifbothdc  3357  ifcldcd  3358  fidifsnen  6331  uzin  8505  fzprval  8944  fztpval  8945  expival  9257
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