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Theorem bnj1534 29298
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 2574 . . 3  |-  F/_ x A
3 nfcv 2574 . . 3  |-  F/_ z A
4 nfv 1630 . . 3  |-  F/ z ( F `  x
)  =/=  ( H `
 x )
5 bnj1534.2 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
65nfcii 2565 . . . . 5  |-  F/_ x F
7 nfcv 2574 . . . . 5  |-  F/_ x
z
86, 7nffv 5738 . . . 4  |-  F/_ x
( F `  z
)
9 nfcv 2574 . . . 4  |-  F/_ x
( H `  z
)
108, 9nfne 2697 . . 3  |-  F/ x
( F `  z
)  =/=  ( H `
 z )
11 fveq2 5731 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
12 fveq2 5731 . . . 4  |-  ( x  =  z  ->  ( H `  x )  =  ( H `  z ) )
1311, 12neeq12d 2618 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  =/=  ( H `
 x )  <->  ( F `  z )  =/=  ( H `  z )
) )
142, 3, 4, 10, 13cbvrab 2956 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  =  {
z  e.  A  | 
( F `  z
)  =/=  ( H `
 z ) }
151, 14eqtri 2458 1  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711   ` cfv 5457
This theorem is referenced by:  bnj1523  29514
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-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465
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