Theorem modc 1940

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/yph

Assertion

Ref	Expression
modc	|- (DECID E.xph -> (E.yA.x(ph -> x = y) <-> A.xA.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/yph
2	1	mo23 1938	. 2 |- (E.yA.x(ph -> x = y) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
3		exmiddc 743	. . 3 |- (DECID E.xph -> (E.xph \/ -. E.xph))
4	1	mor 1939	. . . 4 |- (E.xph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
5	1	mo2n 1925	. . . . 5 |- (-. E.xph -> E.yA.x(ph -> x = y))
6	5	a1d 22	. . . 4 |- (-. E.xph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
7	4, 6	jaoi 635	. . 3 |- ((E.xph \/ -. E.xph) -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
8	3, 7	syl 14	. 2 |- (DECID E.xph -> (A.xA.y((ph /\ [y / x]ph) -> x = y) -> E.yA.x(ph -> x = y)))
9	2, 8	impbid2 131	1 |- (DECID E.xph -> (E.yA.x(ph -> x = y) <-> A.xA.y((ph /\ [y / x]ph) -> x = y)))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628  DECID wdc 741  A.wal 1240   F/wnf 1346  E.wex 1378  [wsb 1642

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643

This theorem is referenced by:  mo2dc  1952