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Theorem ralrab2 2844
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 2423 . . 3  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
21raleqi 2628 . 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 2843 . 2  |-  ( A. x  e.  { y  |  ( y  e.  A  /\  ph ) } ps  <->  A. y ( ( y  e.  A  /\  ph )  ->  ch )
)
5 impexp 261 . . . 4  |-  ( ( ( y  e.  A  /\  ph )  ->  ch ) 
<->  ( y  e.  A  ->  ( ph  ->  ch ) ) )
65albii 1446 . . 3  |-  ( A. y ( ( y  e.  A  /\  ph )  ->  ch )  <->  A. y
( y  e.  A  ->  ( ph  ->  ch ) ) )
7 df-ral 2419 . . 3  |-  ( A. y  e.  A  ( ph  ->  ch )  <->  A. y
( y  e.  A  ->  ( ph  ->  ch ) ) )
86, 7bitr4i 186 . 2  |-  ( A. y ( ( y  e.  A  /\  ph )  ->  ch )  <->  A. y  e.  A  ( ph  ->  ch ) )
92, 4, 83bitri 205 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    /\ wa 103    <-> wb 104   A.wal 1329    e. wcel 1480   {cab 2123   A.wral 2414   {crab 2418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423
This theorem is referenced by: (None)
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