Theorem ifexg 4542

Description: Existence of the conditional operator (closed form). (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		simpl 487	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
2		simpr 489	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
3	1, 2	ifexd 4541	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  Vcvv 3463  ifcif 4492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-if 4493

This theorem is referenced by:  fsuppmptif  9358  cantnfp1lem1  9646  cantnfp1lem3  9648  symgextfv  19487  pmtrfv  19521  marrepeval  22688  gsummatr01lem3  22782  stdbdmetval  24639  stdbdxmet  24640  ellimc2  26004  cdleme31fv  41053