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Theorem bnj1476 28879
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1476.1  |-  D  =  { x  e.  A  |  -.  ph }
bnj1476.2  |-  ( ps 
->  D  =  (/) )
Assertion
Ref Expression
bnj1476  |-  ( ps 
->  A. x  e.  A  ph )

Proof of Theorem bnj1476
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bnj1476.2 . . . 4  |-  ( ps 
->  D  =  (/) )
2 eq0 3469 . . . . 5  |-  ( D  =  (/)  <->  A. y  -.  y  e.  D )
3 bnj1476.1 . . . . . . . . 9  |-  D  =  { x  e.  A  |  -.  ph }
4 nfrab1 2720 . . . . . . . . 9  |-  F/_ x { x  e.  A  |  -.  ph }
53, 4nfcxfr 2416 . . . . . . . 8  |-  F/_ x D
65nfcri 2413 . . . . . . 7  |-  F/ x  y  e.  D
76nfn 1765 . . . . . 6  |-  F/ x  -.  y  e.  D
8 nfv 1605 . . . . . 6  |-  F/ y  -.  x  e.  D
9 eleq1 2343 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  D  <->  x  e.  D ) )
109notbid 285 . . . . . 6  |-  ( y  =  x  ->  ( -.  y  e.  D  <->  -.  x  e.  D ) )
117, 8, 10cbval 1924 . . . . 5  |-  ( A. y  -.  y  e.  D  <->  A. x  -.  x  e.  D )
122, 11bitri 240 . . . 4  |-  ( D  =  (/)  <->  A. x  -.  x  e.  D )
131, 12sylib 188 . . 3  |-  ( ps 
->  A. x  -.  x  e.  D )
143rabeq2i 2785 . . . . . . 7  |-  ( x  e.  D  <->  ( x  e.  A  /\  -.  ph ) )
1514notbii 287 . . . . . 6  |-  ( -.  x  e.  D  <->  -.  (
x  e.  A  /\  -.  ph ) )
1615biimpi 186 . . . . 5  |-  ( -.  x  e.  D  ->  -.  ( x  e.  A  /\  -.  ph ) )
17 iman 413 . . . . 5  |-  ( ( x  e.  A  ->  ph )  <->  -.  ( x  e.  A  /\  -.  ph ) )
1816, 17sylibr 203 . . . 4  |-  ( -.  x  e.  D  -> 
( x  e.  A  ->  ph ) )
1918alimi 1546 . . 3  |-  ( A. x  -.  x  e.  D  ->  A. x ( x  e.  A  ->  ph )
)
2013, 19syl 15 . 2  |-  ( ps 
->  A. x ( x  e.  A  ->  ph )
)
2120bnj1142 28821 1  |-  ( ps 
->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   (/)c0 3455
This theorem is referenced by:  bnj1312  29088
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-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-rab 2552  df-v 2790  df-dif 3155  df-nul 3456
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