Theorem euan 2261

Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Hypothesis

Ref	Expression
moanim.1	⊢ Ⅎxφ

Assertion

Ref	Expression
euan	⊢ (∃!x(φ ∧ ψ) ↔ (φ ∧ ∃!xψ))

Proof of Theorem euan

Step	Hyp	Ref	Expression
1		moanim.1	. . . . . 6 ⊢ Ⅎxφ
2		simpl 443	. . . . . 6 ⊢ ((φ ∧ ψ) → φ)
3	1, 2	exlimi 1803	. . . . 5 ⊢ (∃x(φ ∧ ψ) → φ)
4	3	adantr 451	. . . 4 ⊢ ((∃x(φ ∧ ψ) ∧ ∃*x(φ ∧ ψ)) → φ)
5		simpr 447	. . . . . 6 ⊢ ((φ ∧ ψ) → ψ)
6	5	eximi 1576	. . . . 5 ⊢ (∃x(φ ∧ ψ) → ∃xψ)
7	6	adantr 451	. . . 4 ⊢ ((∃x(φ ∧ ψ) ∧ ∃*x(φ ∧ ψ)) → ∃xψ)
8		nfe1 1732	. . . . . 6 ⊢ Ⅎx∃x(φ ∧ ψ)
9	3	a1d 22	. . . . . . . 8 ⊢ (∃x(φ ∧ ψ) → (ψ → φ))
10	9	ancrd 537	. . . . . . 7 ⊢ (∃x(φ ∧ ψ) → (ψ → (φ ∧ ψ)))
11	5, 10	impbid2 195	. . . . . 6 ⊢ (∃x(φ ∧ ψ) → ((φ ∧ ψ) ↔ ψ))
12	8, 11	mobid 2238	. . . . 5 ⊢ (∃x(φ ∧ ψ) → (∃*x(φ ∧ ψ) ↔ ∃*xψ))
13	12	biimpa 470	. . . 4 ⊢ ((∃x(φ ∧ ψ) ∧ ∃*x(φ ∧ ψ)) → ∃*xψ)
14	4, 7, 13	jca32 521	. . 3 ⊢ ((∃x(φ ∧ ψ) ∧ ∃*x(φ ∧ ψ)) → (φ ∧ (∃xψ ∧ ∃*xψ)))
15		eu5 2242	. . 3 ⊢ (∃!x(φ ∧ ψ) ↔ (∃x(φ ∧ ψ) ∧ ∃*x(φ ∧ ψ)))
16		eu5 2242	. . . 4 ⊢ (∃!xψ ↔ (∃xψ ∧ ∃*xψ))
17	16	anbi2i 675	. . 3 ⊢ ((φ ∧ ∃!xψ) ↔ (φ ∧ (∃xψ ∧ ∃*xψ)))
18	14, 15, 17	3imtr4i 257	. 2 ⊢ (∃!x(φ ∧ ψ) → (φ ∧ ∃!xψ))
19		ibar 490	. . . 4 ⊢ (φ → (ψ ↔ (φ ∧ ψ)))
20	1, 19	eubid 2211	. . 3 ⊢ (φ → (∃!xψ ↔ ∃!x(φ ∧ ψ)))
21	20	biimpa 470	. 2 ⊢ ((φ ∧ ∃!xψ) → ∃!x(φ ∧ ψ))
22	18, 21	impbii 180	1 ⊢ (∃!x(φ ∧ ψ) ↔ (φ ∧ ∃!xψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544  ∃!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:  euanv  2265  2eu7  2290  2eu8  2291