Theorem 2exeu 2281

Description: Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)

Assertion

Ref	Expression
2exeu	⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃!x∃!yφ)

Proof of Theorem 2exeu

Step	Hyp	Ref	Expression
1		eumo 2244	. . . 4 ⊢ (∃!x∃yφ → ∃*x∃yφ)
2		euex 2227	. . . . 5 ⊢ (∃!yφ → ∃yφ)
3	2	moimi 2251	. . . 4 ⊢ (∃*x∃yφ → ∃*x∃!yφ)
4	1, 3	syl 15	. . 3 ⊢ (∃!x∃yφ → ∃*x∃!yφ)
5		2euex 2276	. . 3 ⊢ (∃!y∃xφ → ∃x∃!yφ)
6	4, 5	anim12ci 550	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → (∃x∃!yφ ∧ ∃*x∃!yφ))
7		eu5 2242	. 2 ⊢ (∃!x∃!yφ ↔ (∃x∃!yφ ∧ ∃*x∃!yφ))
8	6, 7	sylibr 203	1 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃!x∃!yφ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205

This proof depends on 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 proof 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  df-mo 2209

This theorem is used by:  2eu1  2284  2eu2  2285  2eu3  2286