Theorem mo 2508

Description: Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)

Hypothesis

Ref	Expression
mo.1	|- F/ y ph

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem mo

Step	Hyp	Ref	Expression
1		mo.1	. . 3 |- F/ y ph
2	1	mo2 2479	. 2 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
3	1	mo3 2507	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
4	2, 3	bitr3i 266	1 |- ( E. y A. x ( ph -> x = y ) <-> 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*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:  bj-snsetex  32951