MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexab2 Structured version   Unicode version

Theorem rexab2 3220
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
rexab2  |-  ( E. x  e.  { y  |  ph } ps  <->  E. 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 rexab2
StepHypRef Expression
1 df-rex 2799 . 2  |-  ( E. x  e.  { y  |  ph } ps  <->  E. x ( x  e. 
{ y  |  ph }  /\  ps ) )
2 nfsab1 2440 . . . 4  |-  F/ y  x  e.  { y  |  ph }
3 nfv 1674 . . . 4  |-  F/ y ps
42, 3nfan 1863 . . 3  |-  F/ y ( x  e.  {
y  |  ph }  /\  ps )
5 nfv 1674 . . 3  |-  F/ x
( ph  /\  ch )
6 eleq1 2521 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  y  e.  { y  |  ph }
) )
7 abid 2438 . . . . 5  |-  ( y  e.  { y  | 
ph }  <->  ph )
86, 7syl6bb 261 . . . 4  |-  ( x  =  y  ->  (
x  e.  { y  |  ph }  <->  ph ) )
9 ralab2.1 . . . 4  |-  ( x  =  y  ->  ( ps 
<->  ch ) )
108, 9anbi12d 710 . . 3  |-  ( x  =  y  ->  (
( x  e.  {
y  |  ph }  /\  ps )  <->  ( ph  /\ 
ch ) ) )
114, 5, 10cbvex 1979 . 2  |-  ( E. x ( x  e. 
{ y  |  ph }  /\  ps )  <->  E. y
( ph  /\  ch )
)
121, 11bitri 249 1  |-  ( E. x  e.  { y  |  ph } ps  <->  E. y ( ph  /\  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1587    e. wcel 1758   {cab 2436   E.wrex 2794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-rex 2799
This theorem is referenced by:  rexrab2  3221  tmdgsum2  19780
  Copyright terms: Public domain W3C validator