Theorem euor 2231

Description: Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)

Hypothesis

Ref	Expression
euor.1	⊢ Ⅎxφ

Assertion

Ref	Expression
euor	⊢ ((¬ φ ∧ ∃!xψ) → ∃!x(φ ∨ ψ))

Proof of Theorem euor

Step	Hyp	Ref	Expression
1		euor.1	. . . 4 ⊢ Ⅎxφ
2	1	nfn 1793	. . 3 ⊢ Ⅎx ¬ φ
3		biorf 394	. . 3 ⊢ (¬ φ → (ψ ↔ (φ ∨ ψ)))
4	2, 3	eubid 2211	. 2 ⊢ (¬ φ → (∃!xψ ↔ ∃!x(φ ∨ ψ)))
5	4	biimpa 470	1 ⊢ ((¬ φ ∧ ∃!xψ) → ∃!x(φ ∨ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  Ⅎwnf 1544  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  euorv  2232