Theorem reueq 2951

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 2518	. 2 |- ( B e. A <-> E. x e. A x = B )
2		moeq 2927	. . . 4 |- E* x x = B
3		mormo 2702	. . . 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 2703	. . 3 |- ( E! x e. A x = B <-> ( E. x e. A x = B /\ E* x e. A x = B ) )
6	4, 5	mpbiran2 943	. 2 |- ( E! x e. A x = B <-> E. x e. A x = B )
7	1, 6	bitr4i 187	1 |- ( B e. A <-> E! x e. A x = B )

Syntax hints:   <-> wb 105   = wceq 1364   E*wmo 2039   e. wcel 2160   E.wrex 2469   E!wreu 2470   E*wrmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-rex 2474  df-reu 2475  df-rmo 2476  df-v 2754

This theorem is referenced by:  divfnzn  9653  icoshftf1o  10023