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

Theorem exbidh 1578
Description: Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
exbidh.1  |-  ( ph  ->  A. x ph )
exbidh.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
exbidh  |-  ( ph  ->  ( E. x ps  <->  E. x ch ) )

Proof of Theorem exbidh
StepHypRef Expression
1 exbidh.1 . . 3  |-  ( ph  ->  A. x ph )
2 exbidh.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2alrimih 1552 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 exbi 1568 . 2  |-  ( A. x ( ps  <->  ch )  ->  ( E. x ps  <->  E. x ch ) )
53, 4syl 15 1  |-  ( ph  ->  ( E. x ps  <->  E. x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528
This theorem is referenced by:  exbidv  1612  exbid  1753  drex2  1908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544
This theorem depends on definitions:  df-bi 177  df-ex 1529
  Copyright terms: Public domain W3C validator