Theorem eumo 2058

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 2030	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
3	1, 2	sylibr 134	1 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1492  ∃!weu 2026  ∃*wmo 2027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-mo 2030

This theorem is referenced by:  eumoi  2059  eu5  2073  euimmo  2093  moaneu  2102  eupick  2105  2eumo  2114  moeq3dc  2914  nfunsn  5550  fnoprabg  5976  uptx  13777  txcn  13778