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Theorem ralbinrald 28079
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 452 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( ps 
<->  th ) )
41, 3rspcdv 2900 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  ->  th )
)
5 ralbinrald.2 . . . . . 6  |-  ( x  e.  A  ->  x  =  X )
62bicomd 192 . . . . . 6  |-  ( x  =  X  ->  ( th 
<->  ps ) )
75, 6syl 15 . . . . 5  |-  ( x  e.  A  ->  ( th 
<->  ps ) )
87adantl 452 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( th 
<->  ps ) )
98biimpd 198 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( th  ->  ps ) )
109ralrimdva 2646 . 2  |-  ( ph  ->  ( th  ->  A. x  e.  A  ps )
)
114, 10impbid 183 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556
This theorem is referenced by:  dfdfat2  28098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803
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