Theorem eupick 2658

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 2603	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		mopick 2650	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
3	1, 2	sylan 592	1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812  ∃*wmo 2562  ∃!weu 2593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2564  df-eu 2594

This theorem is used by:  eupicka  2659  eupickb  2660  reupick  4275  reupick3  4276  eusv2nf  5360  reusv2lem3  5365  copsexgw  5466  copsexgwOLD  5467  copsexg  5468  funssres  6577  oprabidw  7444  oprabid  7445  txcn  23852  isch3  31722  bnj849  35434  iotasbc  45243