Theorem rmoeq 3635

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 3604	. . 3 ⊢ ∃*𝑥 𝑥 = 𝐴
2	1	moani 2553	. 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)
3		df-rmo 3061	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴))
4	2, 3	mpbir 234	1 ⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴

Syntax hints:   ∧ wa 399   = wceq 1542   ∈ wcel 2113  ∃*wmo 2538  ∃*wrmo 3056

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 1916  ax-6 1974  ax-7 2019  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-mo 2540  df-cleq 2730  df-rmo 3061

This theorem is referenced by:  nbusgredgeu  27300