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Theorem bnj919 28308
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj919.1  |-  ( ch  <->  ( n  e.  D  /\  F  Fn  n  /\  ph 
/\  ps ) )
bnj919.2  |-  ( ph'  <->  [. P  /  n ]. ph )
bnj919.3  |-  ( ps'  <->  [. P  /  n ]. ps )
bnj919.4  |-  ( ch'  <->  [. P  /  n ]. ch )
bnj919.5  |-  P  e. 
_V
Assertion
Ref Expression
bnj919  |-  ( ch'  <->  ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' ) )
Distinct variable groups:    D, n    n, F    P, n
Allowed substitution hints:    ph( n)    ps( n)    ch( n)    ph'( n)    ps'( n)    ch'( n)

Proof of Theorem bnj919
StepHypRef Expression
1 bnj919.4 . 2  |-  ( ch'  <->  [. P  /  n ]. ch )
2 bnj919.1 . . 3  |-  ( ch  <->  ( n  e.  D  /\  F  Fn  n  /\  ph 
/\  ps ) )
3 bnj919.5 . . 3  |-  P  e. 
_V
42, 3bnj524 28277 . 2  |-  ( [. P  /  n ]. ch  <->  [. P  /  n ]. ( n  e.  D  /\  F  Fn  n  /\  ph  /\  ps )
)
5 df-bnj17 28223 . . . . 5  |-  ( ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' )  <->  ( ( P  e.  D  /\  F  Fn  P  /\  ph' )  /\  ps' ) )
6 nfv 1610 . . . . . . 7  |-  F/ n  P  e.  D
7 nfv 1610 . . . . . . 7  |-  F/ n  F  Fn  P
8 bnj919.2 . . . . . . . 8  |-  ( ph'  <->  [. P  /  n ]. ph )
9 nfsbc1v 3044 . . . . . . . 8  |-  F/ n [. P  /  n ]. ph
108, 9nfxfr 1561 . . . . . . 7  |-  F/ n ph'
116, 7, 10nf3an 1803 . . . . . 6  |-  F/ n
( P  e.  D  /\  F  Fn  P  /\  ph' )
12 bnj919.3 . . . . . . 7  |-  ( ps'  <->  [. P  /  n ]. ps )
13 nfsbc1v 3044 . . . . . . 7  |-  F/ n [. P  /  n ]. ps
1412, 13nfxfr 1561 . . . . . 6  |-  F/ n ps'
1511, 14nfan 1800 . . . . 5  |-  F/ n
( ( P  e.  D  /\  F  Fn  P  /\  ph' )  /\  ps' )
165, 15nfxfr 1561 . . . 4  |-  F/ n
( P  e.  D  /\  F  Fn  P  /\  ph'  /\  ps' )
17 eleq1 2376 . . . . . 6  |-  ( n  =  P  ->  (
n  e.  D  <->  P  e.  D ) )
18 fneq2 5371 . . . . . . 7  |-  ( n  =  P  ->  ( F  Fn  n  <->  F  Fn  P ) )
19 sbceq1a 3035 . . . . . . . 8  |-  ( n  =  P  ->  ( ph 
<-> 
[. P  /  n ]. ph ) )
2019, 8syl6bbr 254 . . . . . . 7  |-  ( n  =  P  ->  ( ph 
<->  ph' ) )
21 sbceq1a 3035 . . . . . . . 8  |-  ( n  =  P  ->  ( ps 
<-> 
[. P  /  n ]. ps ) )
2221, 12syl6bbr 254 . . . . . . 7  |-  ( n  =  P  ->  ( ps 
<->  ps' ) )
2318, 20, 223anbi123d 1252 . . . . . 6  |-  ( n  =  P  ->  (
( F  Fn  n  /\  ph  /\  ps )  <->  ( F  Fn  P  /\  ph' 
/\  ps' ) ) )
2417, 23anbi12d 691 . . . . 5  |-  ( n  =  P  ->  (
( n  e.  D  /\  ( F  Fn  n  /\  ph  /\  ps )
)  <->  ( P  e.  D  /\  ( F  Fn  P  /\  ph'  /\  ps' ) ) ) )
25 bnj252 28239 . . . . 5  |-  ( ( n  e.  D  /\  F  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  ( F  Fn  n  /\  ph 
/\  ps ) ) )
26 bnj252 28239 . . . . 5  |-  ( ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' )  <->  ( P  e.  D  /\  ( F  Fn  P  /\  ph' 
/\  ps' ) ) )
2724, 25, 263bitr4g 279 . . . 4  |-  ( n  =  P  ->  (
( n  e.  D  /\  F  Fn  n  /\  ph  /\  ps )  <->  ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' ) ) )
2816, 27sbciegf 3056 . . 3  |-  ( P  e.  _V  ->  ( [. P  /  n ]. ( n  e.  D  /\  F  Fn  n  /\  ph  /\  ps )  <->  ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' ) ) )
293, 28ax-mp 8 . 2  |-  ( [. P  /  n ]. (
n  e.  D  /\  F  Fn  n  /\  ph 
/\  ps )  <->  ( P  e.  D  /\  F  Fn  P  /\  ph'  /\  ps' ) )
301, 4, 293bitri 262 1  |-  ( ch'  <->  ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   _Vcvv 2822   [.wsbc 3025    Fn wfn 5287    /\ w-bnj17 28222
This theorem is referenced by:  bnj910  28491  bnj999  28500  bnj907  28508
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-v 2824  df-sbc 3026  df-fn 5295  df-bnj17 28223
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