Theorem ifexd 4541

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 3481	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
3		ifexd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	3	elexd 3481	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
5	2, 4	ifcld 4539	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Vcvv 3458  ifcif 4492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-if 4493

This theorem is used by:  ifexg  4542  evlslem3  22261  mhpsclcl  22340  psgnfzto1stlem  33444  mplmulmvr  33953  esplyind  33989  prjspnfv01  43389  prjspner01  43390  prjspner1  43391  sge0val  47113  hsphoival  47326  hspmbllem2  47374