ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifelpwung GIF version

Theorem ifelpwung 4464
Description: Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
Assertion
Ref Expression
ifelpwung ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵))

Proof of Theorem ifelpwung
StepHypRef Expression
1 ifssun 3539 . 2 if(𝜑, 𝐴, 𝐵) ⊆ (𝐴𝐵)
2 unexg 4426 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3 elpw2g 4140 . . 3 ((𝐴𝐵) ∈ V → (if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵) ↔ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴𝐵)))
42, 3syl 14 . 2 ((𝐴𝑉𝐵𝑊) → (if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵) ↔ if(𝜑, 𝐴, 𝐵) ⊆ (𝐴𝐵)))
51, 4mpbiri 167 1 ((𝐴𝑉𝐵𝑊) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2141  Vcvv 2730  cun 3119  wss 3121  ifcif 3525  𝒫 cpw 3564
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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-uni 3795
This theorem is referenced by:  ifelpwund  4465  ifelpwun  4466
  Copyright terms: Public domain W3C validator