Theorem euor 2638

Description: Introduce a disjunct into a unique existential quantifier. For a version requiring disjoint variables, but fewer axioms, see euorv 2639. (Contributed by NM, 21-Oct-2005.)

Hypothesis

Ref	Expression
euor.nf	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem euor

Step	Hyp	Ref	Expression
1		euor.nf	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfn 1877	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
3		biorf 947	. . 3 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓)))
4	2, 3	eubid 2614	. 2 ⊢ (¬ 𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥(𝜑 ∨ 𝜓)))
5	4	biimpa 480	1 ⊢ ((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑 ∨ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858  Ⅎwnf 1803  ∃!weu 2595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1800  df-nf 1804  df-mo 2566  df-eu 2596

This theorem is referenced by: (None)