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Theorem ralab2 2930
Description: Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ralab2  |-  ( A. x  e.  { y  |  ph } ps  <->  A. y
( ph  ->  ch )
)
Distinct variable groups:    x, y    ch, x    ph, x    ps, y
Allowed substitution hints:    ph( y)    ps( x)    ch( y)

Proof of Theorem ralab2
StepHypRef Expression
1 df-ral 2548 . 2  |-  ( A. x  e.  { y  |  ph } ps  <->  A. x
( x  e.  {
y  |  ph }  ->  ps ) )
2 nfsab1 2273 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1605 . . . 4  |-  F/ y ps
42, 3nfim 1769 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  ->  ps )
5 nfv 1605 . . 3  |-  F/ x
( ph  ->  ch )
6 eleq1 2343 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2271 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7syl6bb 252 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9imbi12d 311 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  ->  ps )  <->  ( ph  ->  ch ) ) )
114, 5, 10cbval 1924 . 2  |-  ( A. x ( x  e. 
{ y  |  ph }  ->  ps )  <->  A. y
( ph  ->  ch )
)
121, 11bitri 240 1  |-  ( A. x  e.  { y  |  ph } ps  <->  A. y
( ph  ->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543
This theorem is referenced by:  ralrab2  2931  ssintab  3879  efgval  15026  efger  15027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548
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