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Theorem 4exdistr 1852
Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
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 anass 630 . . . . . . . 8  |-  ( ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  ( ph  /\  ( ps  /\  ( ch  /\  th ) ) ) )
21exbii 1569 . . . . . . 7  |-  ( E. w ( ( ph  /\ 
ps )  /\  ( ch  /\  th ) )  <->  E. w ( ph  /\  ( ps  /\  ( ch  /\  th ) ) ) )
3 19.42v 1846 . . . . . . . 8  |-  ( E. w ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  <->  ( ph  /\  E. w ( ps  /\  ( ch  /\  th )
) ) )
4 19.42v 1846 . . . . . . . . 9  |-  ( E. w ( ps  /\  ( ch  /\  th )
)  <->  ( ps  /\  E. w ( ch  /\  th ) ) )
54anbi2i 675 . . . . . . . 8  |-  ( (
ph  /\  E. w
( ps  /\  ( ch  /\  th ) ) )  <->  ( ph  /\  ( ps  /\  E. w
( ch  /\  th ) ) ) )
6 19.42v 1846 . . . . . . . . . 10  |-  ( E. w ( ch  /\  th )  <->  ( ch  /\  E. w th ) )
76anbi2i 675 . . . . . . . . 9  |-  ( ( ps  /\  E. w
( ch  /\  th ) )  <->  ( ps  /\  ( ch  /\  E. w th ) ) )
87anbi2i 675 . . . . . . . 8  |-  ( (
ph  /\  ( ps  /\ 
E. w ( ch 
/\  th ) ) )  <-> 
( ph  /\  ( ps  /\  ( ch  /\  E. w th ) ) ) )
93, 5, 83bitri 262 . . . . . . 7  |-  ( E. w ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  <->  ( ph  /\  ( ps  /\  ( ch  /\  E. w th ) ) ) )
102, 9bitri 240 . . . . . 6  |-  ( E. w ( ( ph  /\ 
ps )  /\  ( ch  /\  th ) )  <-> 
( ph  /\  ( ps  /\  ( ch  /\  E. w th ) ) ) )
1110exbii 1569 . . . . 5  |-  ( E. z E. w ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  E. z
( ph  /\  ( ps  /\  ( ch  /\  E. w th ) ) ) )
12 19.42v 1846 . . . . 5  |-  ( E. z ( ph  /\  ( ps  /\  ( ch  /\  E. w th ) ) )  <->  ( ph  /\ 
E. z ( ps 
/\  ( ch  /\  E. w th ) ) ) )
13 19.42v 1846 . . . . . 6  |-  ( E. z ( ps  /\  ( ch  /\  E. w th ) )  <->  ( ps  /\ 
E. z ( ch 
/\  E. w th )
) )
1413anbi2i 675 . . . . 5  |-  ( (
ph  /\  E. z
( ps  /\  ( ch  /\  E. w th ) ) )  <->  ( ph  /\  ( ps  /\  E. z ( ch  /\  E. w th ) ) ) )
1511, 12, 143bitri 262 . . . 4  |-  ( E. z E. w ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  ( ph  /\  ( ps  /\  E. z ( ch  /\  E. w th ) ) ) )
1615exbii 1569 . . 3  |-  ( E. y E. z E. w ( ( ph  /\ 
ps )  /\  ( ch  /\  th ) )  <->  E. y ( ph  /\  ( ps  /\  E. z
( ch  /\  E. w th ) ) ) )
17 19.42v 1846 . . 3  |-  ( E. y ( ph  /\  ( ps  /\  E. z
( ch  /\  E. w th ) ) )  <-> 
( ph  /\  E. y
( ps  /\  E. z ( ch  /\  E. w th ) ) ) )
1816, 17bitri 240 . 2  |-  ( E. y E. z E. w ( ( ph  /\ 
ps )  /\  ( ch  /\  th ) )  <-> 
( ph  /\  E. y
( ps  /\  E. z ( ch  /\  E. w th ) ) ) )
1918exbii 1569 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 set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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