Theorem eupick 1055

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, which apparently does not appear explicitly in the literature, can be quite useful because it lets us eliminate existential quantifiers in a hypothesis.

Assertion

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

Proof of Theorem eupick

Step	Hyp	Ref	Expression
1		mopick 1054	. 2 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))
2		eumo 1037	. 2 ⊢ (∃!xφ → ∃*xφ)
3	1, 2	sylan 343	1 ⊢ ((∃!xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678  ∃!weu 1007  ∃*wmo 1008

This theorem is referenced by:  eupickb 1056  reupick 1578  funssres 2698  tz6.12-1 2842  chcmh 5148

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