Theorem ifexg 4190

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 4123	. . 3 ⊢ (𝑥 = 𝐴 → if(𝜑, 𝑥, 𝑦) = if(𝜑, 𝐴, 𝑦))
2	1	eleq1d 2715	. 2 ⊢ (𝑥 = 𝐴 → (if(𝜑, 𝑥, 𝑦) ∈ V ↔ if(𝜑, 𝐴, 𝑦) ∈ V))
3		ifeq2 4124	. . 3 ⊢ (𝑦 = 𝐵 → if(𝜑, 𝐴, 𝑦) = if(𝜑, 𝐴, 𝐵))
4	3	eleq1d 2715	. 2 ⊢ (𝑦 = 𝐵 → (if(𝜑, 𝐴, 𝑦) ∈ V ↔ if(𝜑, 𝐴, 𝐵) ∈ V))
5		vex 3234	. . 3 ⊢ 𝑥 ∈ V
6		vex 3234	. . 3 ⊢ 𝑦 ∈ V
7	5, 6	ifex 4189	. 2 ⊢ if(𝜑, 𝑥, 𝑦) ∈ V
8	2, 4, 7	vtocl2g 3301	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → if(𝜑, 𝐴, 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ifcif 4119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-un 3612  df-if 4120

This theorem is referenced by:  fsuppmptif  8346  cantnfp1lem1  8613  cantnfp1lem3  8615  symgextfv  17884  pmtrfv  17918  evlslem3  19562  marrepeval  20417  gsummatr01lem3  20511  stdbdmetval  22366  stdbdxmet  22367  ellimc2  23686  psgnfzto1stlem  29978  cdleme31fv  35995  sge0val  40901  hsphoival  41114  hspmbllem2  41162