Theorem eu2 2509

Description: An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)

Hypothesis

Ref	Expression
eu2.1	|- F/ y ph

Assertion

Ref	Expression
eu2	|- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem eu2

Step	Hyp	Ref	Expression
1		eu5 2496	. 2 |- ( E! x ph <-> ( E. x ph /\ E* x ph ) )
2		eu2.1	. . . 4 |- F/ y ph
3	2	mo3 2507	. . 3 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
4	3	anbi2i 730	. 2 |- ( ( E. x ph /\ E* x ph ) <-> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
5	1, 4	bitri 264	1 |- ( E! x ph <-> ( E. x ph /\ A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   [wsb 1880   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:  reu2  3394  bnj1321  31095