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Theorem ralbinrald 27844
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 453 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( ps 
<->  th ) )
41, 3rspcdv 3015 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  ->  th )
)
5 ralbinrald.2 . . . . . 6  |-  ( x  e.  A  ->  x  =  X )
62bicomd 193 . . . . . 6  |-  ( x  =  X  ->  ( th 
<->  ps ) )
75, 6syl 16 . . . . 5  |-  ( x  e.  A  ->  ( th 
<->  ps ) )
87adantl 453 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( th 
<->  ps ) )
98biimpd 199 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( th  ->  ps ) )
109ralrimdva 2756 . 2  |-  ( ph  ->  ( th  ->  A. x  e.  A  ps )
)
114, 10impbid 184 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666
This theorem is referenced by:  dfdfat2  27862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918
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