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Theorem ifelpwun 4477
Description: Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)
Hypotheses
Ref Expression
ifelpwun.1 𝐴 ∈ V
ifelpwun.2 𝐵 ∈ V
Assertion
Ref Expression
ifelpwun if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵)

Proof of Theorem ifelpwun
StepHypRef Expression
1 ifelpwun.1 . 2 𝐴 ∈ V
2 ifelpwun.2 . 2 𝐵 ∈ V
3 ifelpwung 4475 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵))
41, 2, 3mp2an 426 1 if(𝜑, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wcel 2146  Vcvv 2735  cun 3125  ifcif 3532  𝒫 cpw 3572
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-rab 2462  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-uni 3806
This theorem is referenced by:  fmelpw1o  14118  bj-charfun  14119
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