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Theorem reusv2lem1 4604
Description: Lemma for reusv2 4609. (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 3741 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 nfra1 2769 . . . . 5  |-  F/ y A. y  e.  A  x  =  B
32nfmo 2316 . . . 4  |-  F/ y E* x A. y  e.  A  x  =  B
4 rsp 2754 . . . . . . 7  |-  ( A. y  e.  A  x  =  B  ->  ( y  e.  A  ->  x  =  B ) )
54com12 32 . . . . . 6  |-  ( y  e.  A  ->  ( A. y  e.  A  x  =  B  ->  x  =  B ) )
65alrimiv 1773 . . . . 5  |-  ( y  e.  A  ->  A. x
( A. y  e.  A  x  =  B  ->  x  =  B ) )
7 moeq 3214 . . . . 5  |-  E* x  x  =  B
8 moim 2348 . . . . 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, 8mpisyl 21 . . . 4  |-  ( y  e.  A  ->  E* x A. y  e.  A  x  =  B )
103, 9exlimi 1995 . . 3  |-  ( E. y  y  e.  A  ->  E* x A. y  e.  A  x  =  B )
111, 10sylbi 199 . 2  |-  ( A  =/=  (/)  ->  E* x A. y  e.  A  x  =  B )
12 eu5 2325 . . 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 917 . 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 17 1  |-  ( A  =/=  (/)  ->  ( E! x A. y  e.  A  x  =  B  <->  E. x A. y  e.  A  x  =  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887   E!weu 2299   E*wmo 2300    =/= wne 2622   A.wral 2737   (/)c0 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-v 3047  df-dif 3407  df-nul 3732
This theorem is referenced by: (None)
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