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Theorem 4exdistr 1848
Description: Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
Assertion
Ref Expression
4exdistr  |-  ( E. x E. y E. z E. w ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  E. x
( ph  /\  E. y
( ps  /\  E. z ( ch  /\  E. w th ) ) ) )
Distinct variable groups:    ph, y    ph, z    ph, w    ps, z    ps, w    ch, w
Allowed substitution hints:    ph( x)    ps( x, y)    ch( x, y, z)    th( x, y, z, w)

Proof of Theorem 4exdistr
StepHypRef Expression
1 19.42v 1842 . . . . 5  |-  ( E. w ( ch  /\  th )  <->  ( ch  /\  E. w th ) )
21anbi2i 708 . . . 4  |-  ( ( ( ph  /\  ps )  /\  E. w ( ch  /\  th )
)  <->  ( ( ph  /\ 
ps )  /\  ( ch  /\  E. w th ) ) )
3 19.42v 1842 . . . 4  |-  ( E. w ( ( ph  /\ 
ps )  /\  ( ch  /\  th ) )  <-> 
( ( ph  /\  ps )  /\  E. w
( ch  /\  th ) ) )
4 df-3an 1009 . . . 4  |-  ( (
ph  /\  ps  /\  ( ch  /\  E. w th ) )  <->  ( ( ph  /\  ps )  /\  ( ch  /\  E. w th ) ) )
52, 3, 43bitr4i 285 . . 3  |-  ( E. w ( ( ph  /\ 
ps )  /\  ( ch  /\  th ) )  <-> 
( ph  /\  ps  /\  ( ch  /\  E. w th ) ) )
653exbii 1728 . 2  |-  ( E. x E. y E. z E. w ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  E. x E. y E. z (
ph  /\  ps  /\  ( ch  /\  E. w th ) ) )
7 3exdistr 1847 . 2  |-  ( E. x E. y E. z ( ph  /\  ps  /\  ( ch  /\  E. w th ) )  <->  E. x ( ph  /\  E. y ( ps  /\  E. z ( ch  /\  E. w th ) ) ) )
86, 7bitri 257 1  |-  ( E. x E. y E. z E. w ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  E. x
( ph  /\  E. y
( ps  /\  E. z ( ch  /\  E. w th ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-ex 1672
This theorem is referenced by: (None)
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