Theorem eupick 2628

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 2571	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		mopick 2620	. 2 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))
3	1, 2	sylan 580	1 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1781  ∃*wmo 2531  ∃!weu 2561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-mo 2533  df-eu 2562

This theorem is referenced by:  eupicka  2629  eupickb  2630  reupick  4283  reupick3  4284  eusv2nf  5355  reusv2lem3  5360  copsexgw  5452  copsexg  5453  funssres  6550  oprabidw  7393  oprabid  7394  txcn  23014  isch3  30246  bnj849  33626  iotasbc  42821