Theorem eqeuel 3941

Description: A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022.)

Assertion

Ref	Expression
eqeuel	|- ( ( A =/= (/) /\ A. x A. y ( ( x e. A /\ y e. A ) -> x = y ) ) -> E! x x e. A )

Distinct variable group:   x, y, A

Proof of Theorem eqeuel

Step	Hyp	Ref	Expression
1		n0 3931	. . . 4 |- ( A =/= (/) <-> E. x x e. A )
2	1	biimpi 206	. . 3 |- ( A =/= (/) -> E. x x e. A )
3	2	anim1i 592	. 2 |- ( ( A =/= (/) /\ A. x A. y ( ( x e. A /\ y e. A ) -> x = y ) ) -> ( E. x x e. A /\ A. x A. y ( ( x e. A /\ y e. A ) -> x = y ) ) )
4		eleq1w 2684	. . 3 |- ( x = y -> ( x e. A <-> y e. A ) )
5	4	eu4 2518	. 2 |- ( E! x x e. A <-> ( E. x x e. A /\ A. x A. y ( ( x e. A /\ y e. A ) -> x = y ) ) )
6	3, 5	sylibr 224	1 |- ( ( A =/= (/) /\ A. x A. y ( ( x e. A /\ y e. A ) -> x = y ) ) -> E! x x e. A )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   E!weu 2470   =/= wne 2794   (/)c0 3915

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-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  frgr2wwlk1  27193