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Theorem bnj1534 28930
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 2540 . . 3  |-  F/_ x A
3 nfcv 2540 . . 3  |-  F/_ z A
4 nfv 1626 . . 3  |-  F/ z ( F `  x
)  =/=  ( H `
 x )
5 bnj1534.2 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
65nfcii 2531 . . . . 5  |-  F/_ x F
7 nfcv 2540 . . . . 5  |-  F/_ x
z
86, 7nffv 5694 . . . 4  |-  F/_ x
( F `  z
)
9 nfcv 2540 . . . 4  |-  F/_ x
( H `  z
)
108, 9nfne 2658 . . 3  |-  F/ x
( F `  z
)  =/=  ( H `
 z )
11 fveq2 5687 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
12 fveq2 5687 . . . 4  |-  ( x  =  z  ->  ( H `  x )  =  ( H `  z ) )
1311, 12neeq12d 2582 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  =/=  ( H `
 x )  <->  ( F `  z )  =/=  ( H `  z )
) )
142, 3, 4, 10, 13cbvrab 2914 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  =  {
z  e.  A  | 
( F `  z
)  =/=  ( H `
 z ) }
151, 14eqtri 2424 1  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   ` cfv 5413
This theorem is referenced by:  bnj1523  29146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421
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