Theorem 2eumo 2668

Description: Nested unique existential quantifier and at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)

Assertion

Ref	Expression
2eumo	⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)

Proof of Theorem 2eumo

Step	Hyp	Ref	Expression
1		euimmo 2642	. 2 ⊢ (∀𝑥(∃!𝑦𝜑 → ∃*𝑦𝜑) → (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑))
2		eumo 2604	. 2 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑)
3	1, 2	mpg 1816	1 ⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)

Syntax hints:   → wi 4  ∃*wmo 2563  ∃!weu 2594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-mo 2565  df-eu 2595

This theorem is referenced by: (None)