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Theorem rexrab2 2933
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab2.1  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rexrab2  |-  ( E. x  e.  { y  e.  A  |  ph } ps  <->  E. y  e.  A  ( ph  /\  ch )
)
Distinct variable groups:    x, y    x, A    ch, x    ph, x    ps, y
Allowed substitution hints:    ph( y)    ps( x)    ch( y)    A( y)

Proof of Theorem rexrab2
StepHypRef Expression
1 df-rab 2552 . . 3  |-  { y  e.  A  |  ph }  =  { y  |  ( y  e.  A  /\  ph ) }
21rexeqi 2741 . 2  |-  ( E. x  e.  { y  e.  A  |  ph } ps  <->  E. x  e.  {
y  |  ( y  e.  A  /\  ph ) } ps )
3 ralab2.1 . . 3  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
43rexab2 2932 . 2  |-  ( E. x  e.  { y  |  ( y  e.  A  /\  ph ) } ps  <->  E. y ( ( y  e.  A  /\  ph )  /\  ch )
)
5 anass 630 . . . 4  |-  ( ( ( y  e.  A  /\  ph )  /\  ch ) 
<->  ( y  e.  A  /\  ( ph  /\  ch ) ) )
65exbii 1569 . . 3  |-  ( E. y ( ( y  e.  A  /\  ph )  /\  ch )  <->  E. y
( y  e.  A  /\  ( ph  /\  ch ) ) )
7 df-rex 2549 . . 3  |-  ( E. y  e.  A  (
ph  /\  ch )  <->  E. y ( y  e.  A  /\  ( ph  /\ 
ch ) ) )
86, 7bitr4i 243 . 2  |-  ( E. y ( ( y  e.  A  /\  ph )  /\  ch )  <->  E. y  e.  A  ( ph  /\ 
ch ) )
92, 4, 83bitri 262 1  |-  ( E. x  e.  { y  e.  A  |  ph } ps  <->  E. y  e.  A  ( ph  /\  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   {cab 2269   E.wrex 2544   {crab 2547
This theorem is referenced by:  frminex  4373  sstotbnd3  26500
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-or 359  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-nfc 2408  df-rex 2549  df-rab 2552
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