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Theorem mo2dc 1937
 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 1925 . 2 (DECID xφ → (yx(φx = y) ↔ xy((φ [y / x]φ) → x = y)))
31nfri 1393 . . 3 (φyφ)
43mo3h 1935 . 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 733  ∀wal 1226  Ⅎwnf 1329  ∃wex 1362  [wsb 1627  ∃*wmo 1883 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410 This theorem depends on definitions:  df-bi 110  df-dc 734  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886 This theorem is referenced by: (None)
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