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Theorem eeeanv 1914
 Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
eeeanv (xyz(φ ψ χ) ↔ (xφ yψ zχ))
Distinct variable groups:   φ,y   φ,z   x,z,ψ   x,y,χ
Allowed substitution hints:   φ(x)   ψ(y)   χ(z)

Proof of Theorem eeeanv
StepHypRef Expression
1 df-3an 936 . . 3 ((φ ψ χ) ↔ ((φ ψ) χ))
213exbii 1584 . 2 (xyz(φ ψ χ) ↔ xyz((φ ψ) χ))
3 eeanv 1913 . . 3 (yz((φ ψ) χ) ↔ (y(φ ψ) zχ))
43exbii 1582 . 2 (xyz((φ ψ) χ) ↔ x(y(φ ψ) zχ))
5 eeanv 1913 . . . 4 (xy(φ ψ) ↔ (xφ yψ))
65anbi1i 676 . . 3 ((xy(φ ψ) zχ) ↔ ((xφ yψ) zχ))
7 19.41v 1901 . . 3 (x(y(φ ψ) zχ) ↔ (xy(φ ψ) zχ))
8 df-3an 936 . . 3 ((xφ yψ zχ) ↔ ((xφ yψ) zχ))
96, 7, 83bitr4i 268 . 2 (x(y(φ ψ) zχ) ↔ (xφ yψ zχ))
102, 4, 93bitri 262 1 (xyz(φ ψ χ) ↔ (xφ yψ zχ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545 This theorem is referenced by:  vtocl3  2911  spc3egv  2943  eloprabga  5578  mucass  6135  taddc  6229  letc  6231
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