Theorem eu3h 1986

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 1971	. . 3 |- ( E! x ph -> E. x ph )
2		eu3h.1	. . . 4 |- ( ph -> A. y ph )
3	2	eumo0 1972	. . 3 |- ( E! x ph -> E. y A. x ( ph -> x = y ) )
4	1, 3	jca 300	. 2 |- ( E! x ph -> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
5	2	nfi 1391	. . . . 5 |- F/ y ph
6	5	mo23 1982	. . . 4 |- ( E. y A. x ( ph -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
7	6	anim2i 334	. . 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 1985	. . 3 |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
9	7, 8	sylibr 132	. 2 |- ( ( E. x ph /\ E. y A. x ( ph -> x = y ) ) -> E! x ph )
10	4, 9	impbii 124	1 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   [wsb 1685   E!weu 1941

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

This theorem is referenced by:  eu3  1987  mo2r  1993  2eu4  2034