Theorem ifexg 4107

Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011.)

Assertion

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

Proof of Theorem ifexg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ifeq1 4040	. . 3 ⊢ (𝑥 = 𝐴 → if(𝜑, 𝑥, 𝑦) = if(𝜑, 𝐴, 𝑦))
2	1	eleq1d 2672	. 2 ⊢ (𝑥 = 𝐴 → (if(𝜑, 𝑥, 𝑦) ∈ V ↔ if(𝜑, 𝐴, 𝑦) ∈ V))
3		ifeq2 4041	. . 3 ⊢ (𝑦 = 𝐵 → if(𝜑, 𝐴, 𝑦) = if(𝜑, 𝐴, 𝐵))
4	3	eleq1d 2672	. 2 ⊢ (𝑦 = 𝐵 → (if(𝜑, 𝐴, 𝑦) ∈ V ↔ if(𝜑, 𝐴, 𝐵) ∈ V))
5		vex 3176	. . 3 ⊢ 𝑥 ∈ V
6		vex 3176	. . 3 ⊢ 𝑦 ∈ V
7	5, 6	ifex 4106	. 2 ⊢ if(𝜑, 𝑥, 𝑦) ∈ V
8	2, 4, 7	vtocl2g 3243	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037

This theorem is referenced by:  fsuppmptif  8188  cantnfp1lem1  8458  cantnfp1lem3  8460  symgextfv  17661  pmtrfv  17695  evlslem3  19335  marrepeval  20188  gsummatr01lem3  20282  stdbdmetval  22129  stdbdxmet  22130  ellimc2  23447  psgnfzto1stlem  29181  cdleme31fv  34696  sge0val  39259  hsphoival  39469  hspmbllem2  39517