Theorem eu3h 2044

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 2029	. . 3 |- ( E! x ph -> E. x ph )
2		eu3h.1	. . . 4 |- ( ph -> A. y ph )
3	2	eumo0 2030	. . 3 |- ( E! x ph -> E. y A. x ( ph -> x = y ) )
4	1, 3	jca 304	. 2 |- ( E! x ph -> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
5	2	nfi 1438	. . . . 5 |- F/ y ph
6	5	mo23 2040	. . . 4 |- ( E. y A. x ( ph -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
7	6	anim2i 339	. . 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 2043	. . 3 |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
9	7, 8	sylibr 133	. 2 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> E! x ph )
10	4, 9	impbii 125	1 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   E.wex 1468   [wsb 1735   E!weu 1999

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002

This theorem is referenced by:  eu3  2045  mo2r  2051  2eu4  2092