MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2reu5lem1 Unicode version

Theorem 2reu5lem1 3107
Description: Lemma for 2reu5 3110. Note that  E! x  e.  A E! y  e.  B ph does not mean "there is exactly one  x in  A and exactly one  y in  B such that  ph holds;" see comment for 2eu5 2346. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2reu5lem1  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x E! y ( x  e.  A  /\  y  e.  B  /\  ph )
)
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem 2reu5lem1
StepHypRef Expression
1 df-reu 2681 . . 3  |-  ( E! y  e.  B  ph  <->  E! y ( y  e.  B  /\  ph )
)
21reubii 2862 . 2  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x  e.  A  E! y
( y  e.  B  /\  ph ) )
3 df-reu 2681 . . 3  |-  ( E! x  e.  A  E! y ( y  e.  B  /\  ph )  <->  E! x ( x  e.  A  /\  E! y ( y  e.  B  /\  ph ) ) )
4 euanv 2323 . . . . . 6  |-  ( E! y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  E! y ( y  e.  B  /\  ph ) ) )
54bicomi 194 . . . . 5  |-  ( ( x  e.  A  /\  E! y ( y  e.  B  /\  ph )
)  <->  E! y ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
6 3anass 940 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  <->  ( x  e.  A  /\  ( y  e.  B  /\  ph ) ) )
76bicomi 194 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  y  e.  B  /\  ph )
)
87eubii 2271 . . . . 5  |-  ( E! y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  E! y
( x  e.  A  /\  y  e.  B  /\  ph ) )
95, 8bitri 241 . . . 4  |-  ( ( x  e.  A  /\  E! y ( y  e.  B  /\  ph )
)  <->  E! y ( x  e.  A  /\  y  e.  B  /\  ph )
)
109eubii 2271 . . 3  |-  ( E! x ( x  e.  A  /\  E! y ( y  e.  B  /\  ph ) )  <->  E! x E! y ( x  e.  A  /\  y  e.  B  /\  ph )
)
113, 10bitri 241 . 2  |-  ( E! x  e.  A  E! y ( y  e.  B  /\  ph )  <->  E! x E! y ( x  e.  A  /\  y  e.  B  /\  ph ) )
122, 11bitri 241 1  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x E! y ( x  e.  A  /\  y  e.  B  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721   E!weu 2262   E!wreu 2676
This theorem is referenced by:  2reu5lem3  3109
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
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-reu 2681
  Copyright terms: Public domain W3C validator