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Theorem bnj1534 28885
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1534.1  |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }
bnj1534.2  |-  ( w  e.  F  ->  A. x  w  e.  F )
Assertion
Ref Expression
bnj1534  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
Distinct variable groups:    w, A, x, z    w, F, z   
w, H, x, z
Allowed substitution hints:    D( x, z, w)    F( x)

Proof of Theorem bnj1534
StepHypRef Expression
1 bnj1534.1 . 2  |-  D  =  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }
2 nfcv 2419 . . 3  |-  F/_ x A
3 nfcv 2419 . . 3  |-  F/_ z A
4 nfv 1605 . . 3  |-  F/ z ( F `  x
)  =/=  ( H `
 x )
5 bnj1534.2 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
65nfcii 2410 . . . . 5  |-  F/_ x F
7 nfcv 2419 . . . . 5  |-  F/_ x
z
86, 7nffv 5532 . . . 4  |-  F/_ x
( F `  z
)
9 nfcv 2419 . . . 4  |-  F/_ x
( H `  z
)
108, 9nfne 2539 . . 3  |-  F/ x
( F `  z
)  =/=  ( H `
 z )
11 fveq2 5525 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
12 fveq2 5525 . . . 4  |-  ( x  =  z  ->  ( H `  x )  =  ( H `  z ) )
1311, 12neeq12d 2461 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  =/=  ( H `
 x )  <->  ( F `  z )  =/=  ( H `  z )
) )
142, 3, 4, 10, 13cbvrab 2786 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  =  {
z  e.  A  | 
( F `  z
)  =/=  ( H `
 z ) }
151, 14eqtri 2303 1  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   ` cfv 5255
This theorem is referenced by:  bnj1523  29101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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