Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  euorv Structured version   Visualization version   GIF version

Theorem euorv 2695
 Description: Introduce a disjunct into a unique existential quantifier. Version of euor 2694 requiring disjoint variables, but fewer axioms. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)
Assertion
Ref Expression
euorv ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem euorv
StepHypRef Expression
1 biorf 933 . . 3 𝜑 → (𝜓 ↔ (𝜑𝜓)))
21eubidv 2671 . 2 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑𝜓)))
32biimpa 479 1 ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843  ∃!weu 2652 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 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-mo 2622  df-eu 2653 This theorem is referenced by:  eueq2  3677
 Copyright terms: Public domain W3C validator