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

Theorem albidh 1577
Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
albidh.1  |-  ( ph  ->  A. x ph )
albidh.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
albidh  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )

Proof of Theorem albidh
StepHypRef Expression
1 albidh.1 . . 3  |-  ( ph  ->  A. x ph )
2 albidh.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2alrimih 1552 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 albi 1551 . 2  |-  ( A. x ( ps  <->  ch )  ->  ( A. x ps  <->  A. x ch ) )
53, 4syl 15 1  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527
This theorem is referenced by:  albidv  1611  albid  1752  ax10lem4  1881  ax9  1889  dral2  1906  dral2-o  2120  ax11indalem  2136  ax11inda2ALT  2137  ax11inda  2139  ax10lem18ALT  29124
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
  Copyright terms: Public domain W3C validator