Theorem eu4 2076

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 2061	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		eu4.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	mo4 2075	. . 3 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
4	3	anbi2i 453	. 2 |- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )
5	1, 4	bitri 183	1 |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   E.wex 1480   E!weu 2014   E*wmo 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by:  euequ1  2109  eueq  2896  euind  2912  eusv1  4429  eroveu  6588  climeu  11233  pceu  12223