Theorem moabex 5413

Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.) Avoid axioms. (Revised by SN, 2-Feb-2026.)

Assertion

Ref	Expression
moabex	⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Proof of Theorem moabex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfmo 2541	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		df-sn 4583	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3		vsnex 5381	. . . . . 6 ⊢ {𝑦} ∈ V
4	2, 3	eqeltrri 2834	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = 𝑦} ∈ V
5	4	a1i 11	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝑥 = 𝑦} ∈ V)
6		ss2abim 4014	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 = 𝑦})
7	5, 6	ssexd 5271	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
8	7	exlimiv 1932	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
9	1, 8	sylbi 217	1 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  {cab 2715  Vcvv 3442  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-un 3908  df-in 3910  df-ss 3920  df-sn 4583  df-pr 4585

This theorem is referenced by:  rmorabex  5415  euabex  5416  satfv0  35571