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Theorem ralrab2 3102
Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ralrab2  |-  ( A. x  e.  { y  e.  A  |  ph } ps 
<-> 
A. y  e.  A  ( ph  ->  ch )
)
Distinct variable groups:    x, y    x, A    ch, x    ph, x    ps, y
Allowed substitution hints:    ph( y)    ps( x)    ch( y)    A( y)

Proof of Theorem ralrab2
StepHypRef Expression
1 df-rab 2716 . . 3  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
21raleqi 2910 . 2  |-  ( A. x  e.  { y  e.  A  |  ph } ps 
<-> 
A. x  e.  {
y  |  ( y  e.  A  /\  ph ) } ps )
3 ralab2.1 . . 3  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
43ralab2 3101 . 2  |-  ( A. x  e.  { y  |  ( y  e.  A  /\  ph ) } ps  <->  A. y ( ( y  e.  A  /\  ph )  ->  ch )
)
5 impexp 435 . . . 4  |-  ( ( ( y  e.  A  /\  ph )  ->  ch ) 
<->  ( y  e.  A  ->  ( ph  ->  ch ) ) )
65albii 1576 . . 3  |-  ( A. y ( ( y  e.  A  /\  ph )  ->  ch )  <->  A. y
( y  e.  A  ->  ( ph  ->  ch ) ) )
7 df-ral 2712 . . 3  |-  ( A. y  e.  A  ( ph  ->  ch )  <->  A. y
( y  e.  A  ->  ( ph  ->  ch ) ) )
86, 7bitr4i 245 . 2  |-  ( A. y ( ( y  e.  A  /\  ph )  ->  ch )  <->  A. y  e.  A  ( ph  ->  ch ) )
92, 4, 83bitri 264 1  |-  ( A. x  e.  { y  e.  A  |  ph } ps 
<-> 
A. y  e.  A  ( ph  ->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    e. wcel 1726   {cab 2424   A.wral 2707   {crab 2711
This theorem is referenced by:  efgsf  15363  ghmcnp  18146  nmogelb  18752  pntlem3  21305  sstotbnd2  26485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716
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