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

Theorem albidh 1643
Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-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 1613 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 albi 1610 . 2  |-  ( A. x ( ps  <->  ch )  ->  ( A. x ps  <->  A. x ch ) )
53, 4syl 16 1  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603
This theorem depends on definitions:  df-bi 185
This theorem is referenced by:  albidv  1680  albid  1821  dral2-o  2238  ax12indalem  2253  ax12inda2ALT  2254  ax12inda  2256
  Copyright terms: Public domain W3C validator