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Theorem 19.42vvvv 1832
Description: Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
Assertion
Ref Expression
19.42vvvv  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  ( ph  /\ 
E. w E. x E. y E. z ps ) )
Distinct variable groups:    ph, w    ph, x    ph, y    ph, z
Allowed substitution hints:    ps( x, y, z, w)

Proof of Theorem 19.42vvvv
StepHypRef Expression
1 19.42vv 1830 . . 3  |-  ( E. y E. z (
ph  /\  ps )  <->  (
ph  /\  E. y E. z ps ) )
212exbii 1538 . 2  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  E. w E. x ( ph  /\  E. y E. z ps ) )
3 19.42vv 1830 . 2  |-  ( E. w E. x (
ph  /\  E. y E. z ps )  <->  ( ph  /\ 
E. w E. x E. y E. z ps ) )
42, 3bitri 182 1  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  ( ph  /\ 
E. w E. x E. y E. z ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  ceqsex8v  2645  enq0tr  6686
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