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Theorem 19.42vvv 2042
Description: Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
Assertion
Ref Expression
19.42vvv  |-  ( E. x E. y E. z ( ph  /\  ps )  <->  ( ph  /\  E. x E. y E. z ps ) )
Distinct variable groups:    ph, x    ph, y    ph, z
Allowed substitution hints:    ps( x, y, z)

Proof of Theorem 19.42vvv
StepHypRef Expression
1 19.42vv 2041 . . 3  |-  ( E. y E. z (
ph  /\  ps )  <->  (
ph  /\  E. y E. z ps ) )
21exbii 1580 . 2  |-  ( E. x E. y E. z ( ph  /\  ps )  <->  E. x ( ph  /\ 
E. y E. z ps ) )
3 19.42v 2039 . 2  |-  ( E. x ( ph  /\  E. y E. z ps )  <->  ( ph  /\  E. x E. y E. z ps ) )
42, 3bitri 242 1  |-  ( E. x E. y E. z ( ph  /\  ps )  <->  ( ph  /\  E. x E. y E. z ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537
This theorem is referenced by:  ceqsex6v  2779
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-17 1628  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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