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Theorem bnj1534 32201
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 2616 . . 3  |-  F/_ x A
3 nfcv 2616 . . 3  |-  F/_ z A
4 nfv 1674 . . 3  |-  F/ z ( F `  x
)  =/=  ( H `
 x )
5 bnj1534.2 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
65nfcii 2606 . . . . 5  |-  F/_ x F
7 nfcv 2616 . . . . 5  |-  F/_ x
z
86, 7nffv 5809 . . . 4  |-  F/_ x
( F `  z
)
9 nfcv 2616 . . . 4  |-  F/_ x
( H `  z
)
108, 9nfne 2783 . . 3  |-  F/ x
( F `  z
)  =/=  ( H `
 z )
11 fveq2 5802 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
12 fveq2 5802 . . . 4  |-  ( x  =  z  ->  ( H `  x )  =  ( H `  z ) )
1311, 12neeq12d 2731 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  =/=  ( H `
 x )  <->  ( F `  z )  =/=  ( H `  z )
) )
142, 3, 4, 10, 13cbvrab 3076 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  =  {
z  e.  A  | 
( F `  z
)  =/=  ( H `
 z ) }
151, 14eqtri 2483 1  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1368    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   ` cfv 5529
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537
This theorem is referenced by:  bnj1523  32417
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