Theorem rmoeq 3405

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	|- E* x e. B x = A

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem rmoeq

Step	Hyp	Ref	Expression
1		moeq 3382	. . 3 |- E* x x = A
2	1	moani 2525	. 2 |- E* x ( x e. B /\ x = A )
3		df-rmo 2920	. 2 |- ( E* x e. B x = A <-> E* x ( x e. B /\ x = A ) )
4	2, 3	mpbir 221	1 |- E* x e. B x = A

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   E*wmo 2471   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-rmo 2920  df-v 3202

This theorem is referenced by:  nbusgredgeu  26268