Theorem reueq 3006

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 2561	. 2 |- ( B e. A <-> E. x e. A x = B )
2		moeq 2982	. . . 4 |- E* x x = B
3		mormo 2751	. . . 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 2752	. . 3 |- ( E! x e. A x = B <-> ( E. x e. A x = B /\ E* x e. A x = B ) )
6	4, 5	mpbiran2 950	. 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 1398   E*wmo 2080   e. wcel 2202   E.wrex 2512   E!wreu 2513   E*wrmo 2514

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-rex 2517  df-reu 2518  df-rmo 2519  df-v 2805

This theorem is referenced by:  divfnzn  9916  icoshftf1o  10287