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Theorem prtlem5 26719
Description: Lemma for prter1 26742, prter2 26744, prter3 26745 and prtex 26743. (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 1630 . 2  |-  F/ v E. x  e.  A  ( r  e.  x  /\  s  e.  x
)
2 elequ1 1729 . . . . 5  |-  ( u  =  r  ->  (
u  e.  x  <->  r  e.  x ) )
3 elequ1 1729 . . . . 5  |-  ( v  =  s  ->  (
v  e.  x  <->  s  e.  x ) )
42, 3bi2anan9r 846 . . . 4  |-  ( ( v  =  s  /\  u  =  r )  ->  ( ( u  e.  x  /\  v  e.  x )  <->  ( r  e.  x  /\  s  e.  x ) ) )
54rexbidv 2728 . . 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 2155 . 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 2152 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 178    /\ wa 360   [wsb 1659   E.wrex 2708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-6 1745  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-rex 2713
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