Theorem rmoeq 3679

Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by NM, 29-Oct-2020.)

Assertion

Ref	Expression
rmoeq	⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rmoeq

Step	Hyp	Ref	Expression
1		moeq 3648	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
2	1	moani 2557	. 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)
3		df-rmo 3344	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
4	2, 3	mpbir 232	1 ⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  ∃*wrmo 3343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-cleq 2731  df-rmo 3344

This theorem is referenced by:  nbusgredgeu  29453