Theorem 2eumo 2107

Description: Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)

Assertion

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

Proof of Theorem 2eumo

Step	Hyp	Ref	Expression
1		euimmo 2086	. 2 ⊢ (∀𝑥(∃!𝑦𝜑 → ∃*𝑦𝜑) → (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑))
2		eumo 2051	. 2 ⊢ (∃!𝑦𝜑 → ∃*𝑦𝜑)
3	1, 2	mpg 1444	1 ⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)

Syntax hints:   → wi 4  ∃!weu 2019  ∃*wmo 2020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023

This theorem is referenced by: (None)