Theorem eupick 2632

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 2577	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		mopick 2624	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
3	1, 2	sylan 580	1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1778  ∃*wmo 2537  ∃!weu 2567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-eu 2568

This theorem is referenced by:  eupicka  2633  eupickb  2634  reupick  4328  reupick3  4329  eusv2nf  5394  reusv2lem3  5399  copsexgw  5494  copsexg  5495  funssres  6609  oprabidw  7463  oprabid  7464  txcn  23635  isch3  31261  bnj849  34940  iotasbc  44443