MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  4exbidv Structured version   Unicode version

Theorem 4exbidv 1640
Description: Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
Hypothesis
Ref Expression
4exbidv.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
4exbidv  |-  ( ph  ->  ( E. x E. y E. z E. w ps 
<->  E. x E. y E. z E. w ch ) )
Distinct variable groups:    ph, x    ph, y    ph, z    ph, w
Allowed substitution hints:    ps( x, y, z, w)    ch( x, y, z, w)

Proof of Theorem 4exbidv
StepHypRef Expression
1 4exbidv.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
212exbidv 1638 . 2  |-  ( ph  ->  ( E. z E. w ps  <->  E. z E. w ch ) )
322exbidv 1638 1  |-  ( ph  ->  ( E. x E. y E. z E. w ps 
<->  E. x E. y E. z E. w ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1550
This theorem is referenced by:  ceqsex8v  2997  copsex4g  4445  opbrop  4955  ov3  6210  brecop  6997  th3q  7013  dihopelvalcpre  32046  xihopellsmN  32052  dihopellsm  32053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626
This theorem depends on definitions:  df-bi 178  df-ex 1551
  Copyright terms: Public domain W3C validator