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Theorem ralrab2 2729
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 2332 . . 3  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
21raleqi 2526 . 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 2728 . 2  |-  ( A. x  e.  { y  |  ( y  e.  A  /\  ph ) } ps  <->  A. y ( ( y  e.  A  /\  ph )  ->  ch )
)
5 impexp 254 . . . 4  |-  ( ( ( y  e.  A  /\  ph )  ->  ch ) 
<->  ( y  e.  A  ->  ( ph  ->  ch ) ) )
65albii 1375 . . 3  |-  ( A. y ( ( y  e.  A  /\  ph )  ->  ch )  <->  A. y
( y  e.  A  ->  ( ph  ->  ch ) ) )
7 df-ral 2328 . . 3  |-  ( A. y  e.  A  ( ph  ->  ch )  <->  A. y
( y  e.  A  ->  ( ph  ->  ch ) ) )
86, 7bitr4i 180 . 2  |-  ( A. y ( ( y  e.  A  /\  ph )  ->  ch )  <->  A. y  e.  A  ( ph  ->  ch ) )
92, 4, 83bitri 199 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 101    <-> wb 102   A.wal 1257    e. wcel 1409   {cab 2042   A.wral 2323   {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332
This theorem is referenced by: (None)
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