ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eumo0 Structured version   GIF version

Theorem eumo0 1909
Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
Hypothesis
Ref Expression
eumo0.1 (φyφ)
Assertion
Ref Expression
eumo0 (∃!xφyx(φx = y))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem eumo0
StepHypRef Expression
1 eumo0.1 . . 3 (φyφ)
21euf 1883 . 2 (∃!xφyx(φx = y))
3 bi1 111 . . . 4 ((φx = y) → (φx = y))
43alimi 1320 . . 3 (x(φx = y) → x(φx = y))
54eximi 1469 . 2 (yx(φx = y) → yx(φx = y))
62, 5sylbi 114 1 (∃!xφyx(φx = y))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1224  wex 1358  ∃!weu 1878
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-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406
This theorem depends on definitions:  df-bi 110  df-eu 1881
This theorem is referenced by:  eu2  1922  eu3h  1923
  Copyright terms: Public domain W3C validator