Theorem rmorabex 5440

Description: Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)

Assertion

Ref	Expression
rmorabex	⊢ (∃*𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Proof of Theorem rmorabex

Step	Hyp	Ref	Expression
1		moabex 5439	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
2		df-rmo 3364	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 3421	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	eleq1i 2826	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
5	1, 2, 4	3imtr4i 292	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∃*wmo 2538  {cab 2714  ∃*wrmo 3363  {crab 3420  Vcvv 3464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-rmo 3364  df-rab 3421  df-v 3466  df-un 3936  df-in 3938  df-ss 3948  df-sn 4607  df-pr 4609

This theorem is referenced by:  supexd  9470