Theorem eumo 2046

Description: Existential uniqueness implies "at most one". (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)

Assertion

Ref	Expression
eumo	⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)

Proof of Theorem eumo

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2		df-mo 2018	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 133	1 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1480  ∃!weu 2014  ∃*wmo 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-mo 2018

This theorem is referenced by:  eumoi  2047  eu5  2061  euimmo  2081  moaneu  2090  eupick  2093  2eumo  2102  moeq3dc  2902  nfunsn  5520  fnoprabg  5943  uptx  12914  txcn  12915