MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexcom4a Unicode version

Theorem rexcom4a 2808
Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
Assertion
Ref Expression
rexcom4a  |-  ( E. x E. y  e.  A  ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
Distinct variable groups:    x, A    x, y    ph, x
Allowed substitution hints:    ph( y)    ps( x, y)    A( y)

Proof of Theorem rexcom4a
StepHypRef Expression
1 rexcom4 2807 . 2  |-  ( E. y  e.  A  E. x ( ph  /\  ps )  <->  E. x E. y  e.  A  ( ph  /\ 
ps ) )
2 19.42v 1846 . . 3  |-  ( E. x ( ph  /\  ps )  <->  ( ph  /\  E. x ps ) )
32rexbii 2568 . 2  |-  ( E. y  e.  A  E. x ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
41, 3bitr3i 242 1  |-  ( E. x E. y  e.  A  ( ph  /\  ps )  <->  E. y  e.  A  ( ph  /\  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528   E.wrex 2544
This theorem is referenced by:  rexcom4b  2809  rexcom4aOLD  26344
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-v 2790
  Copyright terms: Public domain W3C validator