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Theorem ifelpwund 4467
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 4466 . 2 ((𝐴𝑉𝐵𝑊) → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵))
41, 2, 3syl2anc 409 1 (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2141  cun 3119  ifcif 3526  𝒫 cpw 3566
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 4107  ax-pr 4194  ax-un 4418
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 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797
This theorem is referenced by:  ifexd  4469
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