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Theorem 2reu5lem1 2970
Description: Lemma for 2reu5 2973. 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 2227. (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 2550 . . 3  |-  ( E! y  e.  B  ph  <->  E! y ( y  e.  B  /\  ph )
)
21reubii 2726 . 2  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x  e.  A  E! y
( y  e.  B  /\  ph ) )
3 df-reu 2550 . . 3  |-  ( E! x  e.  A  E! y ( y  e.  B  /\  ph )  <->  E! x ( x  e.  A  /\  E! y ( y  e.  B  /\  ph ) ) )
4 euanv 2204 . . . . . 6  |-  ( E! y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  E! y ( y  e.  B  /\  ph ) ) )
54bicomi 193 . . . . 5  |-  ( ( x  e.  A  /\  E! y ( y  e.  B  /\  ph )
)  <->  E! y ( x  e.  A  /\  (
y  e.  B  /\  ph ) ) )
6 3anass 938 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  <->  ( x  e.  A  /\  ( y  e.  B  /\  ph ) ) )
76bicomi 193 . . . . . 6  |-  ( ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  ( x  e.  A  /\  y  e.  B  /\  ph )
)
87eubii 2152 . . . . 5  |-  ( E! y ( x  e.  A  /\  ( y  e.  B  /\  ph ) )  <->  E! y
( x  e.  A  /\  y  e.  B  /\  ph ) )
95, 8bitri 240 . . . 4  |-  ( ( x  e.  A  /\  E! y ( y  e.  B  /\  ph )
)  <->  E! y ( x  e.  A  /\  y  e.  B  /\  ph )
)
109eubii 2152 . . 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 240 . 2  |-  ( E! x  e.  A  E! y ( y  e.  B  /\  ph )  <->  E! x E! y ( x  e.  A  /\  y  e.  B  /\  ph ) )
122, 11bitri 240 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 176    /\ wa 358    /\ w3a 934    e. wcel 1684   E!weu 2143   E!wreu 2545
This theorem is referenced by:  2reu5lem3  2972
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-reu 2550
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