Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1534 Unicode version

Theorem bnj1534 28630
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 2494 . . 3  |-  F/_ x A
3 nfcv 2494 . . 3  |-  F/_ z A
4 nfv 1619 . . 3  |-  F/ z ( F `  x
)  =/=  ( H `
 x )
5 bnj1534.2 . . . . . 6  |-  ( w  e.  F  ->  A. x  w  e.  F )
65nfcii 2485 . . . . 5  |-  F/_ x F
7 nfcv 2494 . . . . 5  |-  F/_ x
z
86, 7nffv 5612 . . . 4  |-  F/_ x
( F `  z
)
9 nfcv 2494 . . . 4  |-  F/_ x
( H `  z
)
108, 9nfne 2615 . . 3  |-  F/ x
( F `  z
)  =/=  ( H `
 z )
11 fveq2 5605 . . . 4  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
12 fveq2 5605 . . . 4  |-  ( x  =  z  ->  ( H `  x )  =  ( H `  z ) )
1311, 12neeq12d 2536 . . 3  |-  ( x  =  z  ->  (
( F `  x
)  =/=  ( H `
 x )  <->  ( F `  z )  =/=  ( H `  z )
) )
142, 3, 4, 10, 13cbvrab 2862 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  =  {
z  e.  A  | 
( F `  z
)  =/=  ( H `
 z ) }
151, 14eqtri 2378 1  |-  D  =  { z  e.  A  |  ( F `  z )  =/=  ( H `  z ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1540    = wceq 1642    e. wcel 1710    =/= wne 2521   {crab 2623   ` cfv 5334
This theorem is referenced by:  bnj1523  28846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342
  Copyright terms: Public domain W3C validator