Theorem mo2dc 2091

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	. . . 4 |- F/ y ph
2	1	nfri 1529	. . 3 |- ( ph -> A. y ph )
3	2	mo3h 2089	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
4	1	modc 2079	. 2 |- (DECID E. x ph -> ( E. y A. x ( ph -> x = y ) <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
5	3, 4	bitr4id 199	1 |- (DECID E. x ph -> ( E* x ph <-> E. y A. x ( ph -> x = y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   A.wal 1361   F/wnf 1470   E.wex 1502   [wsb 1772   E*wmo 2037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040

This theorem is referenced by: (None)