Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  alimdvK Unicode version

Theorem alimdvK 27801
Description: Add universal quantifier to both sides of an equivalence. Does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633. Part of Lemma 5 of [KalishMontague] p. 86. (The other parts are just notbii 289 and imbi12i 318.) (Contributed by NM, 11-Apr-2017.)
Hypothesis
Ref Expression
alimdvK.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
alimdvK  |-  ( ph  ->  ( A. x ps 
->  A. x ch )
)
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem alimdvK
StepHypRef Expression
1 ax-17 1628 . . 3  |-  ( ph  ->  A. x ph )
2 alimdvK.1 . . . 4  |-  ( ph  ->  ( ps  ->  ch ) )
32alimiK 27798 . . 3  |-  ( A. x ph  ->  A. x
( ps  ->  ch ) )
41, 3syl 17 . 2  |-  ( ph  ->  A. x ( ps 
->  ch ) )
5 ax-5 1533 . 2  |-  ( A. x ( ps  ->  ch )  ->  ( A. x ps  ->  A. x ch ) )
64, 5syl 17 1  |-  ( ph  ->  ( A. x ps 
->  A. x ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532
This theorem is referenced by:  albidvK  27803
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-mp 10  ax-5 1533  ax-gen 1536  ax-17 1628
  Copyright terms: Public domain W3C validator