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Theorem albidvK 28147
Description: Add universal quantifier to both sides of an equivalence. Uses only Tarski's FOL axiom schemes (see description for equidK 28136). (Contributed by NM, 11-Apr-2017.)
Hypothesis
Ref Expression
albidvK.1  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
albidvK  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem albidvK
StepHypRef Expression
1 albidvK.1 . . . 4  |-  ( ph  ->  ( ps  <->  ch )
)
21biimpd 200 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
32alimdvK 28145 . 2  |-  ( ph  ->  ( A. x ps 
->  A. x ch )
)
41biimprd 216 . . 3  |-  ( ph  ->  ( ch  ->  ps ) )
54alimdvK 28145 . 2  |-  ( ph  ->  ( A. x ch 
->  A. x ps )
)
63, 5impbid 185 1  |-  ( ph  ->  ( A. x ps  <->  A. x ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532
This theorem is referenced by:  ax11wdemoK  28176  ax12olem19K  28218
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-17 1628
This theorem depends on definitions:  df-bi 179
  Copyright terms: Public domain W3C validator