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Theorem prtlem5 26595
Description: Lemma for prter1 26618, prter2 26620, prter3 26621 and prtex 26619. (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 1626 . 2  |-  F/ v E. x  e.  A  ( r  e.  x  /\  s  e.  x
)
2 elequ1 1724 . . . . 5  |-  ( u  =  r  ->  (
u  e.  x  <->  r  e.  x ) )
3 elequ1 1724 . . . . 5  |-  ( v  =  s  ->  (
v  e.  x  <->  s  e.  x ) )
42, 3bi2anan9r 845 . . . 4  |-  ( ( v  =  s  /\  u  =  r )  ->  ( ( u  e.  x  /\  v  e.  x )  <->  ( r  e.  x  /\  s  e.  x ) ) )
54rexbidv 2687 . . 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 2086 . 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 2087 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 set class
Syntax hints:    <-> wb 177    /\ wa 359   [wsb 1655   E.wrex 2667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-6 1740  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-rex 2672
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