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Theorem bnj1476 32173
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 3761 . . . . 5  |-  ( D  =  (/)  <->  A. y  -.  y  e.  D )
3 bnj1476.1 . . . . . . . . 9  |-  D  =  { x  e.  A  |  -.  ph }
4 nfrab1 3007 . . . . . . . . 9  |-  F/_ x { x  e.  A  |  -.  ph }
53, 4nfcxfr 2614 . . . . . . . 8  |-  F/_ x D
65nfcri 2609 . . . . . . 7  |-  F/ x  y  e.  D
76nfn 1840 . . . . . 6  |-  F/ x  -.  y  e.  D
8 nfv 1674 . . . . . 6  |-  F/ y  -.  x  e.  D
9 eleq1 2526 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  D  <->  x  e.  D ) )
109notbid 294 . . . . . 6  |-  ( y  =  x  ->  ( -.  y  e.  D  <->  -.  x  e.  D ) )
117, 8, 10cbval 1981 . . . . 5  |-  ( A. y  -.  y  e.  D  <->  A. x  -.  x  e.  D )
122, 11bitri 249 . . . 4  |-  ( D  =  (/)  <->  A. x  -.  x  e.  D )
131, 12sylib 196 . . 3  |-  ( ps 
->  A. x  -.  x  e.  D )
143rabeq2i 3075 . . . . . . 7  |-  ( x  e.  D  <->  ( x  e.  A  /\  -.  ph ) )
1514notbii 296 . . . . . 6  |-  ( -.  x  e.  D  <->  -.  (
x  e.  A  /\  -.  ph ) )
1615biimpi 194 . . . . 5  |-  ( -.  x  e.  D  ->  -.  ( x  e.  A  /\  -.  ph ) )
17 iman 424 . . . . 5  |-  ( ( x  e.  A  ->  ph )  <->  -.  ( x  e.  A  /\  -.  ph ) )
1816, 17sylibr 212 . . . 4  |-  ( -.  x  e.  D  -> 
( x  e.  A  ->  ph ) )
1918alimi 1605 . . 3  |-  ( A. x  -.  x  e.  D  ->  A. x ( x  e.  A  ->  ph )
)
2013, 19syl 16 . 2  |-  ( ps 
->  A. x ( x  e.  A  ->  ph )
)
2120bnj1142 32116 1  |-  ( ps 
->  A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803   (/)c0 3746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rab 2808  df-v 3080  df-dif 3440  df-nul 3747
This theorem is referenced by:  bnj1312  32382
  Copyright terms: Public domain W3C validator