ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mo2r Structured version   GIF version

Theorem mo2r 1933
Description: A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)
Hypothesis
Ref Expression
mo2r.1 yφ
Assertion
Ref Expression
mo2r (yx(φx = y) → ∃*xφ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem mo2r
StepHypRef Expression
1 mo2r.1 . . . . 5 yφ
21nfri 1394 . . . 4 (φyφ)
32eu3h 1926 . . 3 (∃!xφ ↔ (xφ yx(φx = y)))
43simplbi2com 1310 . 2 (yx(φx = y) → (xφ∃!xφ))
5 df-mo 1885 . 2 (∃*xφ ↔ (xφ∃!xφ))
64, 5sylibr 137 1 (yx(φx = y) → ∃*xφ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314  wnf 1328  wex 1361  ∃!weu 1881  ∃*wmo 1882
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-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411
This theorem depends on definitions:  df-bi 110  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885
This theorem is referenced by:  mo2icl  2695  rmo2ilem  2823
  Copyright terms: Public domain W3C validator