Theorem eu3h 2123

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)

Hypothesis

Ref	Expression
eu3h.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
eu3h	|- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem eu3h

Step	Hyp	Ref	Expression
1		euex 2107	. . 3 |- ( E! x ph -> E. x ph )
2		eu3h.1	. . . 4 |- ( ph -> A. y ph )
3	2	eumo0 2108	. . 3 |- ( E! x ph -> E. y A. x ( ph -> x = y ) )
4	1, 3	jca 306	. 2 |- ( E! x ph -> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
5	2	nfi 1508	. . . . 5 |- F/ y ph
6	5	mo23 2119	. . . 4 |- ( E. y A. x ( ph -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
7	6	anim2i 342	. . 3 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
8	5	eu2 2122	. . 3 |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
9	7, 8	sylibr 134	. 2 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> E! x ph )
10	4, 9	impbii 126	1 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   E.wex 1538   [wsb 1808   E!weu 2077

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080

This theorem is referenced by:  eu3  2124  mo2r  2130  2eu4  2171