Theorem mo2dc 2003

Description: Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)

Hypothesis

Ref	Expression
mo2dc.1	|- F/ y ph

Assertion

Ref	Expression
mo2dc	|- (DECID 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 mo2dc

Step	Hyp	Ref	Expression
1		mo2dc.1	. . 3 |- F/ y ph
2	1	modc 1991	. 2 |- (DECID E. x ph -> ( E. y A. x ( ph -> x = y ) <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
3	1	nfri 1457	. . 3 |- ( ph -> A. y ph )
4	3	mo3h 2001	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
5	2, 4	syl6rbbr 197	1 |- (DECID E. x ph -> ( E* x ph <-> E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103  DECID wdc 780   A.wal 1287   F/wnf 1394   E.wex 1426   [wsb 1692   E*wmo 1949

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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952

This theorem is referenced by: (None)