Theorem eupick 1434

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 1433	. 2 ⊢ ((∃*xφ ⋀ ∃x(φ ⋀ ψ)) → (φ → ψ))
2		eumo 1411	. 2 ⊢ (∃!xφ → ∃*xφ)
3	1, 2	sylan 448	1 ⊢ ((∃!xφ ⋀ ∃x(φ ⋀ ψ)) → (φ → ψ))

Syntax hints:   → wi 3   ⋀ wa 223  ∃wex 979  ∃!weu 1380  ∃*wmo 1381

This theorem is referenced by:  eupickb 1435  reupick 2277  funssres 3549  tz6.12-1 3733  chcmh 9101

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383

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