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Theorem 19.41vvvv 1845
Description: Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
Assertion
Ref Expression
19.41vvvv  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  ( E. w E. x E. y E. z ph  /\  ps ) )
Distinct variable groups:    ps, w    ps, x    ps, y    ps, z
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem 19.41vvvv
StepHypRef Expression
1 19.41vvv 1844 . . 3  |-  ( E. x E. y E. z ( ph  /\  ps )  <->  ( E. x E. y E. z ph  /\ 
ps ) )
21exbii 1569 . 2  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  E. w
( E. x E. y E. z ph  /\  ps ) )
3 19.41v 1842 . 2  |-  ( E. w ( E. x E. y E. z ph  /\ 
ps )  <->  ( E. w E. x E. y E. z ph  /\  ps ) )
42, 3bitri 240 1  |-  ( E. w E. x E. y E. z ( ph  /\ 
ps )  <->  ( E. w E. x E. y E. z ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528
This theorem is referenced by:  elo  25041
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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