Theorem reueq 2934

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 2503	. 2 |- ( B e. A <-> E. x e. A x = B )
2		moeq 2910	. . . 4 |- E* x x = B
3		mormo 2686	. . . 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 2687	. . 3 |- ( E! x e. A x = B <-> ( E. x e. A x = B /\ E* x e. A x = B ) )
6	4, 5	mpbiran2 941	. 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 1353   E*wmo 2025   e. wcel 2146   E.wrex 2454   E!wreu 2455   E*wrmo 2456

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-rex 2459  df-reu 2460  df-rmo 2461  df-v 2737

This theorem is referenced by:  divfnzn  9594  icoshftf1o  9962