Theorem eueq 2944

Description: Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)

Assertion

Ref	Expression
eueq	|- ( A e. _V <-> E! x x = A )

Distinct variable group:   x, A

Proof of Theorem eueq

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2225	. . . 4 |- ( ( x = A /\ y = A ) -> x = y )
2	1	gen2 1473	. . 3 |- A. x A. y ( ( x = A /\ y = A ) -> x = y )
3	2	biantru 302	. 2 |- ( E. x x = A <-> ( E. x x = A /\ A. x A. y ( ( x = A /\ y = A ) -> x = y ) ) )
4		isset 2778	. 2 |- ( A e. _V <-> E. x x = A )
5		eqeq1 2212	. . 3 |- ( x = y -> ( x = A <-> y = A ) )
6	5	eu4 2116	. 2 |- ( E! x x = A <-> ( E. x x = A /\ A. x A. y ( ( x = A /\ y = A ) -> x = y ) ) )
7	3, 4, 6	3bitr4i 212	1 |- ( A e. _V <-> E! x x = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1515   E!weu 2054   e. wcel 2176   _Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  eueq1  2945  moeq  2948  mosubt  2950  reuhypd  4519  mptfng  5403  gsum0g  13261  gsumval2  13262  upxp  14777