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Theorem albiiK 27896
Description: Add universal quantifier to both sides of an equivalence. Uses only Tarski's FOL axiom schemes (see description for equidK 27889). Part of Lemma 5 of [KalishMontague] p. 86. (The other parts are just notbii 289 and imbi12i 318.) (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
albiiK.1  |-  ( ph  <->  ps )
Assertion
Ref Expression
albiiK  |-  ( A. x ph  <->  A. x ps )

Proof of Theorem albiiK
StepHypRef Expression
1 albiiK.1 . . . 4  |-  ( ph  <->  ps )
21biimpi 188 . . 3  |-  ( ph  ->  ps )
32alimiK 27895 . 2  |-  ( A. x ph  ->  A. x ps )
41biimpri 199 . . 3  |-  ( ps 
->  ph )
54alimiK 27895 . 2  |-  ( A. x ps  ->  A. x ph )
63, 5impbii 182 1  |-  ( A. x ph  <->  A. x ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1532
This theorem is referenced by:  hbe1wK  27921  equextvK  27977
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536
This theorem depends on definitions:  df-bi 179
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