Theorem modc 1943

Description: Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)

Hypothesis

Ref	Expression
modc.1	|- F/ y ph

Assertion

Ref	Expression
modc	|- (DECID E. x ph -> ( 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 modc

Step	Hyp	Ref	Expression
1		modc.1	. . 3 |- F/ y ph
2	1	mo23 1941	. 2 |- ( E. y A. x ( ph -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
3		exmiddc 744	. . 3 |- (DECID E. x ph -> ( E. x ph \/ -. E. x ph ) )
4	1	mor 1942	. . . 4 |- ( E. x ph -> ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E. y A. x ( ph -> x = y ) ) )
5	1	mo2n 1928	. . . . 5 |- ( -. E. x ph -> E. y A. x ( ph -> x = y ) )
6	5	a1d 22	. . . 4 |- ( -. E. x ph -> ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E. y A. x ( ph -> x = y ) ) )
7	4, 6	jaoi 636	. . 3 |- ( ( E. x ph \/ -. E. x ph ) -> ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E. y A. x ( ph -> x = y ) ) )
8	3, 7	syl 14	. 2 |- (DECID E. x ph -> ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E. y A. x ( ph -> x = y ) ) )
9	2, 8	impbid2 131	1 |- (DECID E. x ph -> ( E. y A. x ( ph -> x = y ) <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629  DECID wdc 742   A.wal 1241   F/wnf 1349   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646

This theorem is referenced by:  mo2dc  1955