Theorem eupick 2267

Description: Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that φ is true, and there is also an x (actually the same one) such that φ and ψ are both true, then φ implies ψ regardless of x. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)

Assertion

Ref	Expression
eupick	⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))

Proof of Theorem eupick

Step	Hyp	Ref	Expression
1		eumo 2244	. 2 ⊢ (∃!xφ → ∃*xφ)
2		mopick 2266	. 2 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))
3	1, 2	sylan 457	1 ⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541  ∃!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:  eupicka  2268  eupickb  2269  reupick  3540  reupick3  3541  copsexg  4608  funssres  5145  oprabid  5551