Theorem ifexg 4530

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 486	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
2		simpr 488	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
3	1, 2	ifexd 4529	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  Vcvv 3454  ifcif 4480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-if 4481

This theorem is referenced by:  fsuppmptif  9345  cantnfp1lem1  9633  cantnfp1lem3  9635  symgextfv  19458  pmtrfv  19492  marrepeval  22623  gsummatr01lem3  22717  stdbdmetval  24574  stdbdxmet  24575  ellimc2  25939  cdleme31fv  41014