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Theorem ralbinrald 27991
Description: Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
Hypotheses
Ref Expression
ralbinrald.1  |-  ( ph  ->  X  e.  A )
ralbinrald.2  |-  ( x  e.  A  ->  x  =  X )
ralbinrald.3  |-  ( x  =  X  ->  ( ps 
<->  th ) )
Assertion
Ref Expression
ralbinrald  |-  ( ph  ->  ( A. x  e.  A  ps  <->  th )
)
Distinct variable groups:    x, X    x, A    ph, x    th, x
Allowed substitution hint:    ps( x)

Proof of Theorem ralbinrald
StepHypRef Expression
1 ralbinrald.1 . . 3  |-  ( ph  ->  X  e.  A )
2 ralbinrald.3 . . . 4  |-  ( x  =  X  ->  ( ps 
<->  th ) )
32adantl 454 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( ps 
<->  th ) )
41, 3rspcdv 3061 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  ->  th )
)
5 ralbinrald.2 . . . . . 6  |-  ( x  e.  A  ->  x  =  X )
62bicomd 194 . . . . . 6  |-  ( x  =  X  ->  ( th 
<->  ps ) )
75, 6syl 16 . . . . 5  |-  ( x  e.  A  ->  ( th 
<->  ps ) )
87adantl 454 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( th 
<->  ps ) )
98biimpd 200 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( th  ->  ps ) )
109ralrimdva 2802 . 2  |-  ( ph  ->  ( th  ->  A. x  e.  A  ps )
)
114, 10impbid 185 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   A.wral 2711
This theorem is referenced by:  dfdfat2  28009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-v 2964
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