MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusv2lem1 Structured version   Unicode version

Theorem reusv2lem1 4657
Description: Lemma for reusv2 4662. (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 3803 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
2 nfra1 2838 . . . . 5  |-  F/ y A. y  e.  A  x  =  B
32nfmo 2302 . . . 4  |-  F/ y E* x A. y  e.  A  x  =  B
4 rsp 2823 . . . . . . 7  |-  ( A. y  e.  A  x  =  B  ->  ( y  e.  A  ->  x  =  B ) )
54com12 31 . . . . . 6  |-  ( y  e.  A  ->  ( A. y  e.  A  x  =  B  ->  x  =  B ) )
65alrimiv 1720 . . . . 5  |-  ( y  e.  A  ->  A. x
( A. y  e.  A  x  =  B  ->  x  =  B ) )
7 moeq 3275 . . . . 5  |-  E* x  x  =  B
8 moim 2340 . . . . 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 18 . . . 4  |-  ( y  e.  A  ->  E* x A. y  e.  A  x  =  B )
103, 9exlimi 1913 . . 3  |-  ( E. y  y  e.  A  ->  E* x A. y  e.  A  x  =  B )
111, 10sylbi 195 . 2  |-  ( A  =/=  (/)  ->  E* x A. y  e.  A  x  =  B )
12 eu5 2311 . . 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 906 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819   E!weu 2283   E*wmo 2284    =/= wne 2652   A.wral 2807   (/)c0 3793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3474  df-nul 3794
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator