Theorem reueq 2789

Description: Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)

Assertion

Ref	Expression
reueq	|- ( B e. A <-> E! x e. A x = B )

Distinct variable groups:   x, A   x, B

Proof of Theorem reueq

Step	Hyp	Ref	Expression
1		risset 2394	. 2 |- ( B e. A <-> E. x e. A x = B )
2		moeq 2767	. . . 4 |- E* x x = B
3		mormo 2565	. . . 4 |- ( E* x x = B -> E* x e. A x = B )
4	2, 3	ax-mp 7	. . 3 |- E* x e. A x = B
5		reu5 2566	. . 3 |- ( E! x e. A x = B <-> ( E. x e. A x = B /\ E* x e. A x = B ) )
6	4, 5	mpbiran2 882	. 2 |- ( E! x e. A x = B <-> E. x e. A x = B )
7	1, 6	bitr4i 185	1 |- ( B e. A <-> E! x e. A x = B )

Syntax hints:   <-> wb 103   = wceq 1284   e. wcel 1433   E*wmo 1942   E.wrex 2349   E!wreu 2350   E*wrmo 2351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-rex 2354  df-reu 2355  df-rmo 2356  df-v 2603

This theorem is referenced by:  divfnzn  8706  icoshftf1o  9013