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Theorem reusv2lem1 4551
Description: Lemma for reusv2 4556. (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 3477 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 nfra1 2606 . . . . 5  |-  F/ y A. y  e.  A  x  =  B
32nfmo 2173 . . . 4  |-  F/ y E* x A. y  e.  A  x  =  B
4 rsp 2616 . . . . . . 7  |-  ( A. y  e.  A  x  =  B  ->  ( y  e.  A  ->  x  =  B ) )
54com12 27 . . . . . 6  |-  ( y  e.  A  ->  ( A. y  e.  A  x  =  B  ->  x  =  B ) )
65alrimiv 1621 . . . . 5  |-  ( y  e.  A  ->  A. x
( A. y  e.  A  x  =  B  ->  x  =  B ) )
7 moeq 2954 . . . . 5  |-  E* x  x  =  B
8 moim 2202 . . . . 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 1366 . . . 4  |-  ( y  e.  A  ->  E* x A. y  e.  A  x  =  B )
103, 9exlimi 1813 . . 3  |-  ( E. y  y  e.  A  ->  E* x A. y  e.  A  x  =  B )
111, 10sylbi 187 . 2  |-  ( A  =/=  (/)  ->  E* x A. y  e.  A  x  =  B )
12 eu5 2194 . . 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 873 . 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 15 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 176   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   E*wmo 2157    =/= wne 2459   A.wral 2556   (/)c0 3468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-v 2803  df-dif 3168  df-nul 3469
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