Theorem 2euswap 2280

Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)

Assertion

Ref	Expression
2euswap	⊢ (∀x∃*yφ → (∃!x∃yφ → ∃!y∃xφ))

Proof of Theorem 2euswap

Step	Hyp	Ref	Expression
1		excomim 1742	. . . 4 ⊢ (∃x∃yφ → ∃y∃xφ)
2	1	a1i 10	. . 3 ⊢ (∀x∃*yφ → (∃x∃yφ → ∃y∃xφ))
3		2moswap 2279	. . 3 ⊢ (∀x∃*yφ → (∃*x∃yφ → ∃*y∃xφ))
4	2, 3	anim12d 546	. 2 ⊢ (∀x∃*yφ → ((∃x∃yφ ∧ ∃*x∃yφ) → (∃y∃xφ ∧ ∃*y∃xφ)))
5		eu5 2242	. 2 ⊢ (∃!x∃yφ ↔ (∃x∃yφ ∧ ∃*x∃yφ))
6		eu5 2242	. 2 ⊢ (∃!y∃xφ ↔ (∃y∃xφ ∧ ∃*y∃xφ))
7	4, 5, 6	3imtr4g 261	1 ⊢ (∀x∃*yφ → (∃!x∃yφ → ∃!y∃xφ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541  ∃!weu 2204  ∃*wmo 2205

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  df-mo 2209

This theorem is referenced by:  euxfr2  3022  2reuswap  3039