Theorem ifexd 4534

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 3476	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
3		ifexd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	3	elexd 3476	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
5	2, 4	ifcld 4532	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3453  ifcif 4485

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 2147  ax-9 2155  ax-ext 2734

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 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-if 4486

This theorem is used by:  ifexg  4535  evlslem3  22302  mhpsclcl  22381  psgnfzto1stlem  33548  mplmulmvr  34057  esplyind  34093  prjspnfv01  43478  prjspner01  43479  prjspner1  43480  sge0val  47202  hsphoival  47415  hspmbllem2  47463