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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 xφ
Assertion
Ref Expression
euan (∃!x(φ ψ) ↔ (φ ∃!xψ))

Proof of Theorem euan
StepHypRef Expression
1 moanim.1 . . . . . 6 xφ
2 simpl 443 . . . . . 6 ((φ ψ) → φ)
31, 2exlimi 1803 . . . . 5 (x(φ ψ) → φ)
43adantr 451 . . . 4 ((x(φ ψ) ∃*x(φ ψ)) → φ)
5 simpr 447 . . . . . 6 ((φ ψ) → ψ)
65eximi 1576 . . . . 5 (x(φ ψ) → xψ)
76adantr 451 . . . 4 ((x(φ ψ) ∃*x(φ ψ)) → xψ)
8 nfe1 1732 . . . . . 6 xx(φ ψ)
93a1d 22 . . . . . . . 8 (x(φ ψ) → (ψφ))
109ancrd 537 . . . . . . 7 (x(φ ψ) → (ψ → (φ ψ)))
115, 10impbid2 195 . . . . . 6 (x(φ ψ) → ((φ ψ) ↔ ψ))
128, 11mobid 2238 . . . . 5 (x(φ ψ) → (∃*x(φ ψ) ↔ ∃*xψ))
1312biimpa 470 . . . 4 ((x(φ ψ) ∃*x(φ ψ)) → ∃*xψ)
144, 7, 13jca32 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))
1716anbi2i 675 . . 3 ((φ ∃!xψ) ↔ (φ (xψ ∃*xψ)))
1814, 15, 173imtr4i 257 . 2 (∃!x(φ ψ) → (φ ∃!xψ))
19 ibar 490 . . . 4 (φ → (ψ ↔ (φ ψ)))
201, 19eubid 2211 . . 3 (φ → (∃!xψ∃!x(φ ψ)))
2120biimpa 470 . 2 ((φ ∃!xψ) → ∃!x(φ ψ))
2218, 21impbii 180 1 (∃!x(φ ψ) ↔ (φ ∃!xψ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544  ∃!weu 2204  ∃*wmo 2205 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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
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