Theorem euimmo 2253

Description: Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)

Assertion

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

Proof of Theorem euimmo

Step	Hyp	Ref	Expression
1		eumo 2244	. 2 ⊢ (∃!xψ → ∃*xψ)
2		moim 2250	. 2 ⊢ (∀x(φ → ψ) → (∃*xψ → ∃*xφ))
3	1, 2	syl5 28	1 ⊢ (∀x(φ → ψ) → (∃!xψ → ∃*xφ))

Syntax hints:   → wi 4  ∀wal 1540  ∃!weu 2204  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  euim  2254  2eumo  2277  moeq3  3014  reuss2  3536