Theorem euorv 1972

Description: Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 23-Mar-1995.)

Assertion

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

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem euorv

Step	Hyp	Ref	Expression
1		ax-17 1462	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	euor 1971	1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∨ wo 662  ∃!weu 1945

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-eu 1948

This theorem is referenced by:  eueq2dc  2779