Theorem eu4 2116

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 2101	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		eu4.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	mo4 2115	. . 3 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
4	3	anbi2i 457	. 2 |- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )
5	1, 4	bitri 184	1 |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   E.wex 1515   E!weu 2054   E*wmo 2055

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

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by:  euequ1  2149  eueq  2944  euind  2960  eusv1  4500  eroveu  6715  climeu  11640  pceu  12651