Theorem eumo 1973

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 5	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
2		df-mo 1945	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 132	1 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1421  ∃!weu 1941  ∃*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-mo 1945

This theorem is referenced by:  eumoi  1974  eu5  1988  euimmo  2008  moaneu  2017  eupick  2020  2eumo  2029  moeq3dc  2768  nfunsn  5228  fnoprabg  5622