Theorem 2eu2ex 2278

Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Assertion

Ref	Expression
2eu2ex	⊢ (∃!x∃!yφ → ∃x∃yφ)

Proof of Theorem 2eu2ex

Step	Hyp	Ref	Expression
1		euex 2227	. 2 ⊢ (∃!x∃!yφ → ∃x∃!yφ)
2		euex 2227	. . 3 ⊢ (∃!yφ → ∃yφ)
3	2	eximi 1576	. 2 ⊢ (∃x∃!yφ → ∃x∃yφ)
4	1, 3	syl 15	1 ⊢ (∃!x∃!yφ → ∃x∃yφ)

Syntax hints:   → wi 4  ∃wex 1541  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  2eu1  2284