Theorem ifexg 4513

Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011.) (Proof shortened by BJ, 1-Sep-2022.)

Assertion

Ref	Expression
ifexg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Proof of Theorem ifexg

Step	Hyp	Ref	Expression
1		elex 3512	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3512	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		ifcl 4510	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → if(𝜑, 𝐴, 𝐵) ∈ V)
4	1, 2, 3	syl2an 597	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  Vcvv 3494  ifcif 4466

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3496  df-if 4467

This theorem is referenced by:  fsuppmptif  8857  cantnfp1lem1  9135  cantnfp1lem3  9137  symgextfv  18540  pmtrfv  18574  evlslem3  20287  marrepeval  21166  gsummatr01lem3  21260  stdbdmetval  23118  stdbdxmet  23119  ellimc2  24469  psgnfzto1stlem  30737  cdleme31fv  37520  sge0val  42642  hsphoival  42855  hspmbllem2  42903