Theorem ifexg 4576

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 484	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
2		simpr 486	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
3	1, 2	ifexd 4575	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  Vcvv 3475  ifcif 4527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-if 4528

This theorem is referenced by:  fsuppmptif  9390  cantnfp1lem1  9669  cantnfp1lem3  9671  symgextfv  19279  pmtrfv  19313  marrepeval  22047  gsummatr01lem3  22141  stdbdmetval  24005  stdbdxmet  24006  ellimc2  25376  cdleme31fv  39199