Theorem rmorabex 5435

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 5433	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
2		df-rmo 3365	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rab 3413	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
4	3	eleq1i 2851	. 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 2145  ∃*wmo 2562  {cab 2738  ∃*wrmo 3364  {crab 3412  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-rmo 3365  df-rab 3413  df-v 3452  df-un 3904  df-in 3906  df-ss 3916  df-sn 4585  df-pr 4587

This theorem is used by:  supexd  9426