Theorem reueq 2887

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 2466	. 2 |- ( B e. A <-> E. x e. A x = B )
2		moeq 2863	. . . 4 |- E* x x = B
3		mormo 2645	. . . 4 |- ( E* x x = B -> E* x e. A x = B )
4	2, 3	ax-mp 5	. . 3 |- E* x e. A x = B
5		reu5 2646	. . 3 |- ( E! x e. A x = B <-> ( E. x e. A x = B /\ E* x e. A x = B ) )
6	4, 5	mpbiran2 926	. 2 |- ( E! x e. A x = B <-> E. x e. A x = B )
7	1, 6	bitr4i 186	1 |- ( B e. A <-> E! x e. A x = B )

Syntax hints:   <-> wb 104   = wceq 1332   e. wcel 1481   E*wmo 2001   E.wrex 2418   E!wreu 2419   E*wrmo 2420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-rex 2423  df-reu 2424  df-rmo 2425  df-v 2691

This theorem is referenced by:  divfnzn  9440  icoshftf1o  9804