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Theorem euimmo 2510
 Description: Uniqueness implies "at most one" 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 2487 . 2 (∃!𝑥𝜓 → ∃*𝑥𝜓)
2 moim 2507 . 2 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
31, 2syl5 33 1 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473  ∃!weu 2458  ∃*wmo 2459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463 This theorem is referenced by:  euim  2511  2eumo  2533  moeq3  3350  reuss2  3866
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