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 3477	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
3		ifexd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	3	elexd 3477	. 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 2147  Vcvv 3454  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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-if 4486

This theorem is used by:  ifexg  4535  evlslem3  22419  mhpsclcl  22498  psgnfzto1stlem  33731  mplmulmvr  34241  esplyind  34277  sge0val  47581  hsphoival  47794  hspmbllem2  47842