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Theorem eeeanv 1805
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 886 . . 3 ((φ ψ χ) ↔ ((φ ψ) χ))
213exbii 1495 . 2 (xyz(φ ψ χ) ↔ xyz((φ ψ) χ))
3 eeanv 1804 . . 3 (yz((φ ψ) χ) ↔ (y(φ ψ) zχ))
43exbii 1493 . 2 (xyz((φ ψ) χ) ↔ x(y(φ ψ) zχ))
5 eeanv 1804 . . . 4 (xy(φ ψ) ↔ (xφ yψ))
65anbi1i 431 . . 3 ((xy(φ ψ) zχ) ↔ ((xφ yψ) zχ))
7 19.41v 1779 . . 3 (x(y(φ ψ) zχ) ↔ (xy(φ ψ) zχ))
8 df-3an 886 . . 3 ((xφ yψ zχ) ↔ ((xφ yψ) zχ))
96, 7, 83bitr4i 201 . 2 (x(y(φ ψ) zχ) ↔ (xφ yψ zχ))
102, 4, 93bitri 195 1 (xyz(φ ψ χ) ↔ (xφ yψ zχ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   w3a 884  wex 1378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347
This theorem is referenced by:  vtocl3  2604  spc3egv  2638  spc3gv  2639  eloprabga  5533  prarloc  6486
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