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Theorem prtlem5 30840
Description: Lemma for prter1 30863, prter2 30865, prter3 30866 and prtex 30864. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
prtlem5  |-  ( [ s  /  v ] [ r  /  u ] E. x  e.  A  ( u  e.  x  /\  v  e.  x
)  <->  E. x  e.  A  ( r  e.  x  /\  s  e.  x
) )
Distinct variable groups:    v, u, x, r    u, s, v, x    u, A, v, x
Allowed substitution hints:    A( s, r)

Proof of Theorem prtlem5
StepHypRef Expression
1 nfv 1712 . 2  |-  F/ v E. x  e.  A  ( r  e.  x  /\  s  e.  x
)
2 elequ1 1826 . . . . 5  |-  ( u  =  r  ->  (
u  e.  x  <->  r  e.  x ) )
3 elequ1 1826 . . . . 5  |-  ( v  =  s  ->  (
v  e.  x  <->  s  e.  x ) )
42, 3bi2anan9r 872 . . . 4  |-  ( ( v  =  s  /\  u  =  r )  ->  ( ( u  e.  x  /\  v  e.  x )  <->  ( r  e.  x  /\  s  e.  x ) ) )
54rexbidv 2965 . . 3  |-  ( ( v  =  s  /\  u  =  r )  ->  ( E. x  e.  A  ( u  e.  x  /\  v  e.  x )  <->  E. x  e.  A  ( r  e.  x  /\  s  e.  x ) ) )
65sbiedv 2154 . 2  |-  ( v  =  s  ->  ( [ r  /  u ] E. x  e.  A  ( u  e.  x  /\  v  e.  x
)  <->  E. x  e.  A  ( r  e.  x  /\  s  e.  x
) ) )
71, 6sbie 2151 1  |-  ( [ s  /  v ] [ r  /  u ] E. x  e.  A  ( u  e.  x  /\  v  e.  x
)  <->  E. x  e.  A  ( r  e.  x  /\  s  e.  x
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367   [wsb 1744   E.wrex 2805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-10 1842  ax-12 1859  ax-13 2004
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1618  df-nf 1622  df-sb 1745  df-rex 2810
This theorem is referenced by: (None)
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