Theorem rmorabex 5431

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 5429	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
2		df-rmo 3368	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 3416	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	eleq1i 2854	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
5	1, 2, 4	3imtr4i 295	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  ∃*wmo 2565  {cab 2741  ∃*wrmo 3367  {crab 3415  Vcvv 3453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-rmo 3368  df-rab 3416  df-v 3455  df-ss 3900  df-in 4093  df-un 4153  df-sn 4588  df-pr 4590

This theorem is used by:  supexd  9455