Theorem ifexd 4579

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 3502	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
3		ifexd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	3	elexd 3502	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
5	2, 4	ifcld 4577	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3478  ifcif 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-if 4532

This theorem is referenced by:  ifexg  4580  evlslem3  22122  mhpsclcl  22169  psgnfzto1stlem  33103  prjspnfv01  42611  prjspner01  42612  prjspner1  42613  sge0val  46322  hsphoival  46535  hspmbllem2  46583