Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1476 Structured version   Unicode version

Theorem bnj1476 29610
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 3720 . . . . 5  |-  ( D  =  (/)  <->  A. y  -.  y  e.  D )
3 bnj1476.1 . . . . . . . . 9  |-  D  =  { x  e.  A  |  -.  ph }
4 nfrab1 2948 . . . . . . . . 9  |-  F/_ x { x  e.  A  |  -.  ph }
53, 4nfcxfr 2567 . . . . . . . 8  |-  F/_ x D
65nfcri 2563 . . . . . . 7  |-  F/ x  y  e.  D
76nfn 1960 . . . . . 6  |-  F/ x  -.  y  e.  D
8 nfv 1755 . . . . . 6  |-  F/ y  -.  x  e.  D
9 eleq1 2494 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  D  <->  x  e.  D ) )
109notbid 295 . . . . . 6  |-  ( y  =  x  ->  ( -.  y  e.  D  <->  -.  x  e.  D ) )
117, 8, 10cbval 2086 . . . . 5  |-  ( A. y  -.  y  e.  D  <->  A. x  -.  x  e.  D )
122, 11bitri 252 . . . 4  |-  ( D  =  (/)  <->  A. x  -.  x  e.  D )
131, 12sylib 199 . . 3  |-  ( ps 
->  A. x  -.  x  e.  D )
143rabeq2i 3019 . . . . . . 7  |-  ( x  e.  D  <->  ( x  e.  A  /\  -.  ph ) )
1514notbii 297 . . . . . 6  |-  ( -.  x  e.  D  <->  -.  (
x  e.  A  /\  -.  ph ) )
1615biimpi 197 . . . . 5  |-  ( -.  x  e.  D  ->  -.  ( x  e.  A  /\  -.  ph ) )
17 iman 425 . . . . 5  |-  ( ( x  e.  A  ->  ph )  <->  -.  ( x  e.  A  /\  -.  ph ) )
1816, 17sylibr 215 . . . 4  |-  ( -.  x  e.  D  -> 
( x  e.  A  ->  ph ) )
1918alimi 1678 . . 3  |-  ( A. x  -.  x  e.  D  ->  A. x ( x  e.  A  ->  ph )
)
2013, 19syl 17 . 2  |-  ( ps 
->  A. x ( x  e.  A  ->  ph )
)
2120bnj1142 29553 1  |-  ( ps 
->  A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435    = wceq 1437    e. wcel 1872   A.wral 2714   {crab 2718   (/)c0 3704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-nul 3705
This theorem is referenced by:  bnj1312  29819
  Copyright terms: Public domain W3C validator