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Theorem 2reu5lem1 3309
Description: Lemma for 2reu5 3312. 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 2392. (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 2821 . . 3  |-  ( E! y  e.  B  ph  <->  E! y ( y  e.  B  /\  ph )
)
21reubii 3048 . 2  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x  e.  A  E! y
( y  e.  B  /\  ph ) )
3 df-reu 2821 . . 3  |-  ( E! x  e.  A  E! y ( y  e.  B  /\  ph )  <->  E! x ( x  e.  A  /\  E! y ( y  e.  B  /\  ph ) ) )
4 euanv 2360 . . . . . 6  |-  ( E! y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  E! y ( y  e.  B  /\  ph ) ) )
54bicomi 202 . . . . 5  |-  ( ( x  e.  A  /\  E! y ( y  e.  B  /\  ph )
)  <->  E! y ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
6 3anass 977 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  <->  ( x  e.  A  /\  ( y  e.  B  /\  ph ) ) )
76bicomi 202 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  y  e.  B  /\  ph )
)
87eubii 2300 . . . . 5  |-  ( E! y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  E! y
( x  e.  A  /\  y  e.  B  /\  ph ) )
95, 8bitri 249 . . . 4  |-  ( ( x  e.  A  /\  E! y ( y  e.  B  /\  ph )
)  <->  E! y ( x  e.  A  /\  y  e.  B  /\  ph )
)
109eubii 2300 . . 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 249 . 2  |-  ( E! x  e.  A  E! y ( y  e.  B  /\  ph )  <->  E! x E! y ( x  e.  A  /\  y  e.  B  /\  ph ) )
122, 11bitri 249 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   E!weu 2275   E!wreu 2816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-eu 2279  df-reu 2821
This theorem is referenced by:  2reu5lem3  3311
  Copyright terms: Public domain W3C validator