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Theorem bnj1476 29156
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 3635 . . . . 5  |-  ( D  =  (/)  <->  A. y  -.  y  e.  D )
3 bnj1476.1 . . . . . . . . 9  |-  D  =  { x  e.  A  |  -.  ph }
4 nfrab1 2881 . . . . . . . . 9  |-  F/_ x { x  e.  A  |  -.  ph }
53, 4nfcxfr 2569 . . . . . . . 8  |-  F/_ x D
65nfcri 2566 . . . . . . 7  |-  F/ x  y  e.  D
76nfn 1811 . . . . . 6  |-  F/ x  -.  y  e.  D
8 nfv 1629 . . . . . 6  |-  F/ y  -.  x  e.  D
9 eleq1 2496 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  D  <->  x  e.  D ) )
109notbid 286 . . . . . 6  |-  ( y  =  x  ->  ( -.  y  e.  D  <->  -.  x  e.  D ) )
117, 8, 10cbval 1982 . . . . 5  |-  ( A. y  -.  y  e.  D  <->  A. x  -.  x  e.  D )
122, 11bitri 241 . . . 4  |-  ( D  =  (/)  <->  A. x  -.  x  e.  D )
131, 12sylib 189 . . 3  |-  ( ps 
->  A. x  -.  x  e.  D )
143rabeq2i 2946 . . . . . . 7  |-  ( x  e.  D  <->  ( x  e.  A  /\  -.  ph ) )
1514notbii 288 . . . . . 6  |-  ( -.  x  e.  D  <->  -.  (
x  e.  A  /\  -.  ph ) )
1615biimpi 187 . . . . 5  |-  ( -.  x  e.  D  ->  -.  ( x  e.  A  /\  -.  ph ) )
17 iman 414 . . . . 5  |-  ( ( x  e.  A  ->  ph )  <->  -.  ( x  e.  A  /\  -.  ph ) )
1816, 17sylibr 204 . . . 4  |-  ( -.  x  e.  D  -> 
( x  e.  A  ->  ph ) )
1918alimi 1568 . . 3  |-  ( A. x  -.  x  e.  D  ->  A. x ( x  e.  A  ->  ph )
)
2013, 19syl 16 . 2  |-  ( ps 
->  A. x ( x  e.  A  ->  ph )
)
2120bnj1142 29098 1  |-  ( ps 
->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2698   {crab 2702   (/)c0 3621
This theorem is referenced by:  bnj1312  29365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rab 2707  df-v 2951  df-dif 3316  df-nul 3622
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