Theorem eupick 1976

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 1929	. 2 ⊢ (∃!xφ → ∃*xφ)
2		mopick 1975	. 2 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))
3	1, 2	sylan 267	1 ⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378  ∃!weu 1897  ∃*wmo 1898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by:  eupicka  1977  eupickb  1978  reupick  3215  reupick3  3216  copsexg  3972  eusv2nf  4154  funssres  4885  oprabid  5480