Theorem eu4 2003

Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)

Hypothesis

Ref	Expression
eu4.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
eu4	|- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )

Distinct variable groups:   x, y   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem eu4

Step	Hyp	Ref	Expression
1		eu5 1988	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		eu4.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	mo4 2002	. . 3 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
4	3	anbi2i 444	. 2 |- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )
5	1, 4	bitri 182	1 |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   E!weu 1941   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  euequ1  2036  eueq  2763  euind  2779  eusv1  4202  eroveu  6220  climeu  10135