Theorem eupick 2428

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 2391	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		mopick 2427	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
3	1, 2	sylan 486	1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 382  ∃wex 1694  ∃!weu 2362  ∃*wmo 2363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983

This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1699  df-eu 2366  df-mo 2367

This theorem is referenced by:  eupicka  2429  eupickb  2430  reupick  3773  reupick3  3774  eusv2nf  4689  reusv2lem3  4696  copsexg  4780  funssres  5729  oprabid  6452  txcn  21140  isch3  27271  bnj849  30095  iotasbc  37524