Theorem modc 2069

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 2067	. 2 |- ( E. y A. x ( ph -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
3		exmiddc 836	. . 3 |- (DECID E. x ph -> ( E. x ph \/ -. E. x ph ) )
4	1	mor 2068	. . . 4 |- ( E. x ph -> ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> E. y A. x ( ph -> x = y ) ) )
5	1	mo2n 2054	. . . . 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 716	. . 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 143	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 104   <-> wb 105   \/ wo 708  DECID wdc 834   A.wal 1351   F/wnf 1460   E.wex 1492   [wsb 1762

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763

This theorem is referenced by:  mo2dc  2081