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Theorem ralab2 3063
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 2675 . 2  |-  ( A. x  e.  { y  |  ph } ps  <->  A. x
( x  e.  {
y  |  ph }  ->  ps ) )
2 nfsab1 2398 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1626 . . . 4  |-  F/ y ps
42, 3nfim 1828 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  ->  ps )
5 nfv 1626 . . 3  |-  F/ x
( ph  ->  ch )
6 eleq1 2468 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2396 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7syl6bb 253 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9imbi12d 312 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  ->  ps )  <->  ( ph  ->  ch ) ) )
114, 5, 10cbval 2040 . 2  |-  ( A. x ( x  e. 
{ y  |  ph }  ->  ps )  <->  A. y
( ph  ->  ch )
)
121, 11bitri 241 1  |-  ( A. x  e.  { y  |  ph } ps  <->  A. y
( ph  ->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    e. wcel 1721   {cab 2394   A.wral 2670
This theorem is referenced by:  ralrab2  3064  ssintab  4031  efgval  15308  efger  15309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-ral 2675
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