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Theorem reusv2lem1 4724
Description: Lemma for reusv2 4729. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reusv2lem1  |-  ( A  =/=  (/)  ->  ( E! x A. y  e.  A  x  =  B  <->  E. x A. y  e.  A  x  =  B )
)
Distinct variable groups:    x, y, A    x, B
Allowed substitution hint:    B( y)

Proof of Theorem reusv2lem1
StepHypRef Expression
1 n0 3637 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 nfra1 2756 . . . . 5  |-  F/ y A. y  e.  A  x  =  B
32nfmo 2298 . . . 4  |-  F/ y E* x A. y  e.  A  x  =  B
4 rsp 2766 . . . . . . 7  |-  ( A. y  e.  A  x  =  B  ->  ( y  e.  A  ->  x  =  B ) )
54com12 29 . . . . . 6  |-  ( y  e.  A  ->  ( A. y  e.  A  x  =  B  ->  x  =  B ) )
65alrimiv 1641 . . . . 5  |-  ( y  e.  A  ->  A. x
( A. y  e.  A  x  =  B  ->  x  =  B ) )
7 moeq 3110 . . . . 5  |-  E* x  x  =  B
8 moim 2327 . . . . 5  |-  ( A. x ( A. y  e.  A  x  =  B  ->  x  =  B )  ->  ( E* x  x  =  B  ->  E* x A. y  e.  A  x  =  B ) )
96, 7, 8ee10 1385 . . . 4  |-  ( y  e.  A  ->  E* x A. y  e.  A  x  =  B )
103, 9exlimi 1821 . . 3  |-  ( E. y  y  e.  A  ->  E* x A. y  e.  A  x  =  B )
111, 10sylbi 188 . 2  |-  ( A  =/=  (/)  ->  E* x A. y  e.  A  x  =  B )
12 eu5 2319 . . 3  |-  ( E! x A. y  e.  A  x  =  B  <-> 
( E. x A. y  e.  A  x  =  B  /\  E* x A. y  e.  A  x  =  B )
)
1312rbaib 874 . 2  |-  ( E* x A. y  e.  A  x  =  B  ->  ( E! x A. y  e.  A  x  =  B  <->  E. x A. y  e.  A  x  =  B )
)
1411, 13syl 16 1  |-  ( A  =/=  (/)  ->  ( E! x A. y  e.  A  x  =  B  <->  E. x A. y  e.  A  x  =  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   E*wmo 2282    =/= wne 2599   A.wral 2705   (/)c0 3628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-nul 3629
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