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Theorem bnj919 29650
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 ) )
32sbcbii 3311 . 2  |-  ( [. P  /  n ]. ch  <->  [. P  /  n ]. ( n  e.  D  /\  F  Fn  n  /\  ph  /\  ps )
)
4 bnj919.5 . . 3  |-  P  e. 
_V
5 df-bnj17 29564 . . . . 5  |-  ( ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' )  <->  ( ( P  e.  D  /\  F  Fn  P  /\  ph' )  /\  ps' ) )
6 nfv 1769 . . . . . . 7  |-  F/ n  P  e.  D
7 nfv 1769 . . . . . . 7  |-  F/ n  F  Fn  P
8 bnj919.2 . . . . . . . 8  |-  ( ph'  <->  [. P  /  n ]. ph )
9 nfsbc1v 3275 . . . . . . . 8  |-  F/ n [. P  /  n ]. ph
108, 9nfxfr 1704 . . . . . . 7  |-  F/ n ph'
116, 7, 10nf3an 2033 . . . . . 6  |-  F/ n
( P  e.  D  /\  F  Fn  P  /\  ph' )
12 bnj919.3 . . . . . . 7  |-  ( ps'  <->  [. P  /  n ]. ps )
13 nfsbc1v 3275 . . . . . . 7  |-  F/ n [. P  /  n ]. ps
1412, 13nfxfr 1704 . . . . . 6  |-  F/ n ps'
1511, 14nfan 2031 . . . . 5  |-  F/ n
( ( P  e.  D  /\  F  Fn  P  /\  ph' )  /\  ps' )
165, 15nfxfr 1704 . . . 4  |-  F/ n
( P  e.  D  /\  F  Fn  P  /\  ph'  /\  ps' )
17 eleq1 2537 . . . . . 6  |-  ( n  =  P  ->  (
n  e.  D  <->  P  e.  D ) )
18 fneq2 5675 . . . . . . 7  |-  ( n  =  P  ->  ( F  Fn  n  <->  F  Fn  P ) )
19 sbceq1a 3266 . . . . . . . 8  |-  ( n  =  P  ->  ( ph 
<-> 
[. P  /  n ]. ph ) )
2019, 8syl6bbr 271 . . . . . . 7  |-  ( n  =  P  ->  ( ph 
<->  ph' ) )
21 sbceq1a 3266 . . . . . . . 8  |-  ( n  =  P  ->  ( ps 
<-> 
[. P  /  n ]. ps ) )
2221, 12syl6bbr 271 . . . . . . 7  |-  ( n  =  P  ->  ( ps 
<->  ps' ) )
2318, 20, 223anbi123d 1365 . . . . . 6  |-  ( n  =  P  ->  (
( F  Fn  n  /\  ph  /\  ps )  <->  ( F  Fn  P  /\  ph' 
/\  ps' ) ) )
2417, 23anbi12d 725 . . . . 5  |-  ( n  =  P  ->  (
( n  e.  D  /\  ( F  Fn  n  /\  ph  /\  ps )
)  <->  ( P  e.  D  /\  ( F  Fn  P  /\  ph'  /\  ps' ) ) ) )
25 bnj252 29580 . . . . 5  |-  ( ( n  e.  D  /\  F  Fn  n  /\  ph 
/\  ps )  <->  ( n  e.  D  /\  ( F  Fn  n  /\  ph 
/\  ps ) ) )
26 bnj252 29580 . . . . 5  |-  ( ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' )  <->  ( P  e.  D  /\  ( F  Fn  P  /\  ph' 
/\  ps' ) ) )
2724, 25, 263bitr4g 296 . . . 4  |-  ( n  =  P  ->  (
( n  e.  D  /\  F  Fn  n  /\  ph  /\  ps )  <->  ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' ) ) )
2816, 27sbciegf 3287 . . 3  |-  ( P  e.  _V  ->  ( [. P  /  n ]. ( n  e.  D  /\  F  Fn  n  /\  ph  /\  ps )  <->  ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' ) ) )
294, 28ax-mp 5 . 2  |-  ( [. P  /  n ]. (
n  e.  D  /\  F  Fn  n  /\  ph 
/\  ps )  <->  ( P  e.  D  /\  F  Fn  P  /\  ph'  /\  ps' ) )
301, 3, 293bitri 279 1  |-  ( ch'  <->  ( P  e.  D  /\  F  Fn  P  /\  ph' 
/\  ps' ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031   [.wsbc 3255    Fn wfn 5584    /\ w-bnj17 29563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-sbc 3256  df-fn 5592  df-bnj17 29564
This theorem is referenced by:  bnj910  29831  bnj999  29840  bnj907  29848
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