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Theorem prtlem5 26228
Description: Lemma for prter1 26253, prter2 26255, prter3 26256 and prtex 26254. (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 1624 . 2  |-  F/ v E. x  e.  A  ( r  e.  x  /\  s  e.  x
)
2 nfv 1624 . . 3  |-  F/ u  v  =  s
3 nfvd 1625 . . 3  |-  ( v  =  s  ->  F/ u E. x  e.  A  ( r  e.  x  /\  s  e.  x
) )
4 elequ1 1718 . . . . . 6  |-  ( u  =  r  ->  (
u  e.  x  <->  r  e.  x ) )
5 elequ1 1718 . . . . . 6  |-  ( v  =  s  ->  (
v  e.  x  <->  s  e.  x ) )
64, 5bi2anan9r 844 . . . . 5  |-  ( ( v  =  s  /\  u  =  r )  ->  ( ( u  e.  x  /\  v  e.  x )  <->  ( r  e.  x  /\  s  e.  x ) ) )
76rexbidv 2649 . . . 4  |-  ( ( v  =  s  /\  u  =  r )  ->  ( E. x  e.  A  ( u  e.  x  /\  v  e.  x )  <->  E. x  e.  A  ( r  e.  x  /\  s  e.  x ) ) )
87ex 423 . . 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 ) ) ) )
92, 3, 8sbied 2049 . 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
) ) )
101, 9sbie 2051 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 176    /\ wa 358    = wceq 1647   [wsb 1653    e. wcel 1715   E.wrex 2629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-rex 2634
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