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Theorem euimmo 2505
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 2482 . 2 (∃!𝑥𝜓 → ∃*𝑥𝜓)
2 moim 2502 . 2 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
31, 2syl5 33 1 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  ∃!weu 2453  ∃*wmo 2454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-eu 2457  df-mo 2458
This theorem is referenced by:  euim  2506  2eumo  2528  moeq3  3345  reuss2  3861
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