Theorem euimmo 1045

Description: Uniqueness implies "at most one" through implication.

Assertion

Ref	Expression
euimmo	⊢ (∀x(φ → ψ) → (∃!xψ → ∃*xφ))

Proof of Theorem euimmo

Step	Hyp	Ref	Expression
1		immo 1043	. 2 ⊢ (∀x(φ → ψ) → (∃*xψ → ∃*xφ))
2		eumo 1037	. 2 ⊢ (∃!xψ → ∃*xψ)
3	1, 2	syl5 22	1 ⊢ (∀x(φ → ψ) → (∃!xψ → ∃*xφ))

Syntax hints:   → wi 2  ∀wal 672  ∃!weu 1007  ∃*wmo 1008

This theorem is referenced by:  2eumo 1062  moeq3 1432  reuss 1577

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

metamath.org