Theorem ifelpwund 4513

Description: Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)

Hypotheses

Ref	Expression
ifelpwund.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ifelpwund.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
ifelpwund	⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))

Proof of Theorem ifelpwund

Step	Hyp	Ref	Expression
1		ifelpwund.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		ifelpwund.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		ifelpwung 4512	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))
4	1, 2, 3	syl2anc 411	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝒫 (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∈ wcel 2164   ∪ cun 3151  ifcif 3557  𝒫 cpw 3601

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836

This theorem is referenced by:  ifexd  4515