Theorem eu3v 2498

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a distinct variable condition on ph. (Revised by Wolf Lammen, 29-May-2019.)

Assertion

Ref	Expression
eu3v	|- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem eu3v

Step	Hyp	Ref	Expression
1		eu5 2496	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		mo2v 2477	. . 3 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
3	2	anbi2i 730	. 2 |- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
4	1, 3	bitri 264	1 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   E!weu 2470   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  eqeu  3377  reu3  3396  eunex  4859  bj-eunex  32799