Theorem eu4 2518

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 2496	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		eu4.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	mo4 2517	. . 3 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
4	3	anbi2i 730	. 2 |- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> x = y ) ) )
5	1, 4	bitri 264	1 |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ ps ) -> 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-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475

This theorem is referenced by:  eueq  3378  euind  3393  eqeuel  3941  uniintsn  4514  eusv1  4860  omeu  7665  eroveu  7842  climeu  14286  pceu  15551  initoeu2lem2  16665  psgneu  17926  gsumval3eu  18305  frgr3vlem2  27138  3vfriswmgrlem  27141  unirep  33507  rlimdmafv  41257