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Theorem eean 1912
 Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
eean.1 yφ
eean.2 xψ
Assertion
Ref Expression
eean (xy(φ ψ) ↔ (xφ yψ))

Proof of Theorem eean
StepHypRef Expression
1 eean.1 . . . 4 yφ
2119.42 1880 . . 3 (y(φ ψ) ↔ (φ yψ))
32exbii 1582 . 2 (xy(φ ψ) ↔ x(φ yψ))
4 eean.2 . . . 4 xψ
54nfex 1843 . . 3 xyψ
6519.41 1879 . 2 (x(φ yψ) ↔ (xφ yψ))
73, 6bitri 240 1 (xy(φ ψ) ↔ (xφ yψ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545 This theorem is referenced by:  eeanv  1913  reean  2777
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