Theorem ifelpwund 4533

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 4532	. 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 2177   ∪ cun 3165  ifcif 3572  𝒫 cpw 3617

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pr 4257  ax-un 4484

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-rab 2494  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-uni 3853

This theorem is referenced by:  ifexd  4535