Theorem ifexd 4507

Description: Existence of the conditional operator (deduction form). (Contributed by SN, 26-Jul-2024.)

Hypotheses

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

Assertion

Ref	Expression
ifexd	⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)

Proof of Theorem ifexd

Step	Hyp	Ref	Expression
1		ifexd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	elexd 3452	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
3		ifexd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	3	elexd 3452	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
5	2, 4	ifcld 4505	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432  ifcif 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-if 4460

This theorem is referenced by:  ifexg  4508  evlslem3  21290  mhpsclcl  21337  psgnfzto1stlem  31367  prjspnfv01  40461  prjspner01  40462  prjspner1  40463  sge0val  43904  hsphoival  44117  hspmbllem2  44165