MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  euimmo Structured version   Visualization version   GIF version

Theorem euimmo 2693
Description: Existential uniqueness implies uniqueness through reverse implication. (Contributed by NM, 22-Apr-1995.)
Assertion
Ref Expression
euimmo (∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))

Proof of Theorem euimmo
StepHypRef Expression
1 eumo 2656 . 2 (∃!𝑥𝜓 → ∃*𝑥𝜓)
2 moim 2619 . 2 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
31, 2syl5 34 1 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  ∃*wmo 2613  ∃!weu 2646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-mo 2615  df-eu 2647
This theorem is referenced by:  euim  2694  euimOLD  2695  2eumo  2720  moeq3  3700  reuss2  4280
  Copyright terms: Public domain W3C validator