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Theorem ralab2 3016
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 2633 . 2  |-  ( A. x  e.  { y  |  ph } ps  <->  A. x
( x  e.  {
y  |  ph }  ->  ps ) )
2 nfsab1 2356 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1624 . . . 4  |-  F/ y ps
42, 3nfim 1820 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  ->  ps )
5 nfv 1624 . . 3  |-  F/ x
( ph  ->  ch )
6 eleq1 2426 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2354 . . . . 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 1997 . 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 1545    = wceq 1647    e. wcel 1715   {cab 2352   A.wral 2628
This theorem is referenced by:  ralrab2  3017  ssintab  3981  efgval  15236  efger  15237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-ral 2633
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