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Theorem prtlem5 29150
Description: Lemma for prter1 29173, prter2 29175, prter3 29176 and prtex 29174. (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 1674 . 2  |-  F/ v E. x  e.  A  ( r  e.  x  /\  s  e.  x
)
2 elequ1 1761 . . . . 5  |-  ( u  =  r  ->  (
u  e.  x  <->  r  e.  x ) )
3 elequ1 1761 . . . . 5  |-  ( v  =  s  ->  (
v  e.  x  <->  s  e.  x ) )
42, 3bi2anan9r 869 . . . 4  |-  ( ( v  =  s  /\  u  =  r )  ->  ( ( u  e.  x  /\  v  e.  x )  <->  ( r  e.  x  /\  s  e.  x ) ) )
54rexbidv 2868 . . 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 2114 . 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 2110 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 369   [wsb 1702   E.wrex 2800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-10 1777  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703  df-rex 2805
This theorem is referenced by: (None)
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