Theorem exmodc 2095

Description: If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)

Assertion

Ref	Expression
exmodc	|- (DECID E. x ph -> ( E. x ph \/ E* x ph ) )

Proof of Theorem exmodc

Step	Hyp	Ref	Expression
1		df-dc 836	. 2 |- (DECID E. x ph <-> ( E. x ph \/ -. E. x ph ) )
2		pm2.21 618	. . . 4 |- ( -. E. x ph -> ( E. x ph -> E! x ph ) )
3		df-mo 2049	. . . 4 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
4	2, 3	sylibr 134	. . 3 |- ( -. E. x ph -> E* x ph )
5	4	orim2i 762	. 2 |- ( ( E. x ph \/ -. E. x ph ) -> ( E. x ph \/ E* x ph ) )
6	1, 5	sylbi 121	1 |- (DECID E. x ph -> ( E. x ph \/ E* x ph ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 709  DECID wdc 835   E.wex 1506   E!weu 2045   E*wmo 2046

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-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836  df-mo 2049

This theorem is referenced by: (None)