Theorem euor 1999

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

Hypothesis

Ref	Expression
euor.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

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

Proof of Theorem euor

Step	Hyp	Ref	Expression
1		euor.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	hbn 1613	. . 3 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
3		biorf 716	. . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4	2, 3	eubidh 1979	. 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓)))
5	4	biimpa 292	1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 680  ∀wal 1310  ∃!weu 1973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-17 1487  ax-ial 1495

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-fal 1318  df-eu 1976

This theorem is referenced by:  euorv  2000