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

Theorem ifelpwund 4479
Description: Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)
Hypotheses
Ref Expression
ifelpwund.1 (𝜑𝐴𝑉)
ifelpwund.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
ifelpwund (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵))

Proof of Theorem ifelpwund
StepHypRef Expression
1 ifelpwund.1 . 2 (𝜑𝐴𝑉)
2 ifelpwund.2 . 2 (𝜑𝐵𝑊)
3 ifelpwung 4478 . 2 ((𝐴𝑉𝐵𝑊) → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵))
41, 2, 3syl2anc 411 1 (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  cun 3127  ifcif 3534  𝒫 cpw 3574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808
This theorem is referenced by:  ifexd  4481
  Copyright terms: Public domain W3C validator