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Theorem ralab2 3203
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 2742 . 2  |-  ( A. x  e.  { y  |  ph } ps  <->  A. x
( x  e.  {
y  |  ph }  ->  ps ) )
2 nfsab1 2441 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1761 . . . 4  |-  F/ y ps
42, 3nfim 2003 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  ->  ps )
5 nfv 1761 . . 3  |-  F/ x
( ph  ->  ch )
6 eleq1 2517 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2439 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7syl6bb 265 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9imbi12d 322 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  ->  ps )  <->  ( ph  ->  ch ) ) )
114, 5, 10cbval 2114 . 2  |-  ( A. x ( x  e. 
{ y  |  ph }  ->  ps )  <->  A. y
( ph  ->  ch )
)
121, 11bitri 253 1  |-  ( A. x  e.  { y  |  ph } ps  <->  A. y
( ph  ->  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1442    e. wcel 1887   {cab 2437   A.wral 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-ral 2742
This theorem is referenced by:  ralrab2  3204  ssintab  4251  efgval  17367  efger  17368  elintabg  36180  elintima  36245
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