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Theorem mo2dc 1952
Description: Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
Hypothesis
Ref Expression
mo2dc.1 yφ
Assertion
Ref Expression
mo2dc (DECID xφ → (∃*xφyx(φx = y)))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem mo2dc
StepHypRef Expression
1 mo2dc.1 . . 3 yφ
21modc 1940 . 2 (DECID xφ → (yx(φx = y) ↔ xy((φ [y / x]φ) → x = y)))
31nfri 1409 . . 3 (φyφ)
43mo3h 1950 . 2 (∃*xφxy((φ [y / x]φ) → x = y))
52, 4syl6rbbr 188 1 (DECID xφ → (∃*xφyx(φx = y)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  DECID wdc 741  wal 1240  wnf 1346  wex 1378  [wsb 1642  ∃*wmo 1898
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-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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  df-eu 1900  df-mo 1901
This theorem is referenced by: (None)
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