Theorem eupick 1432

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 1431	. 2 ⊢ ((∃*xφ ⋀ ∃x(φ ⋀ ψ)) → (φ → ψ))
2		eumo 1409	. 2 ⊢ (∃!xφ → ∃*xφ)
3	1, 2	sylan 448	1 ⊢ ((∃!xφ ⋀ ∃x(φ ⋀ ψ)) → (φ → ψ))

Syntax hints:   → wi 3   ⋀ wa 223  ∃wex 978  ∃!weu 1378  ∃*wmo 1379

This theorem is referenced by:  eupickb 1433  reupick 2275  funssres 3544  tz6.12-1 3727  chcmh 9052

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381

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