Theorem reueq 3404

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 3062	. 2 |- ( B e. A <-> E. x e. A x = B )
2		moeq 3382	. . . 4 |- E* x x = B
3		mormo 3158	. . . 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 3159	. . 3 |- ( E! x e. A x = B <-> ( E. x e. A x = B /\ E* x e. A x = B ) )
6	4, 5	mpbiran2 954	. 2 |- ( E! x e. A x = B <-> E. x e. A x = B )
7	1, 6	bitr4i 267	1 |- ( B e. A <-> E! x e. A x = B )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   E*wmo 2471   E.wrex 2913   E!wreu 2914   E*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202

This theorem is referenced by:  icoshftf1o  12295