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Theorem ralab2 3101
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 2712 . 2  |-  ( A. x  e.  { y  |  ph } ps  <->  A. x
( x  e.  {
y  |  ph }  ->  ps ) )
2 nfsab1 2428 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1630 . . . 4  |-  F/ y ps
42, 3nfim 1833 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  ->  ps )
5 nfv 1630 . . 3  |-  F/ x
( ph  ->  ch )
6 eleq1 2498 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2426 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7syl6bb 254 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9imbi12d 313 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  ->  ps )  <->  ( ph  ->  ch ) ) )
114, 5, 10cbval 1983 . 2  |-  ( A. x ( x  e. 
{ y  |  ph }  ->  ps )  <->  A. y
( ph  ->  ch )
)
121, 11bitri 242 1  |-  ( A. x  e.  { y  |  ph } ps  <->  A. y
( ph  ->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550    e. wcel 1726   {cab 2424   A.wral 2707
This theorem is referenced by:  ralrab2  3102  ssintab  4069  efgval  15354  efger  15355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ral 2712
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