Theorem exmodc 2023

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 803	. 2 |- (DECID E. x ph <-> ( E. x ph \/ -. E. x ph ) )
2		pm2.21 589	. . . 4 |- ( -. E. x ph -> ( E. x ph -> E! x ph ) )
3		df-mo 1977	. . . 4 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
4	2, 3	sylibr 133	. . 3 |- ( -. E. x ph -> E* x ph )
5	4	orim2i 733	. 2 |- ( ( E. x ph \/ -. E. x ph ) -> ( E. x ph \/ E* x ph ) )
6	1, 5	sylbi 120	1 |- (DECID E. x ph -> ( E. x ph \/ E* x ph ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 680  DECID wdc 802   E.wex 1449   E!weu 1973   E*wmo 1974

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 587  ax-io 681

This theorem depends on definitions:  df-bi 116  df-dc 803  df-mo 1977

This theorem is referenced by: (None)