Theorem eupick 2630

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 2575	. 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑)
2		mopick 2622	. 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 1780  ∃*wmo 2535  ∃!weu 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2182

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-mo 2537  df-eu 2566

This theorem is referenced by:  eupicka  2631  eupickb  2632  reupick  4278  reupick3  4279  eusv2nf  5335  reusv2lem3  5340  copsexgw  5433  copsexg  5434  funssres  6530  oprabidw  7383  oprabid  7384  txcn  23542  isch3  31223  bnj849  34958  iotasbc  44537