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Theorem bnj976 29601
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj976.1  |-  ( ch  <->  ( N  e.  D  /\  f  Fn  N  /\  ph 
/\  ps ) )
bnj976.2  |-  ( ph'  <->  [. G  /  f ]. ph )
bnj976.3  |-  ( ps'  <->  [. G  /  f ]. ps )
bnj976.4  |-  ( ch'  <->  [. G  /  f ]. ch )
bnj976.5  |-  G  e. 
_V
Assertion
Ref Expression
bnj976  |-  ( ch'  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
Distinct variable groups:    D, f    f, N
Allowed substitution hints:    ph( f)    ps( f)    ch( f)    G( f)    ph'( f)    ps'( f)    ch'( f)

Proof of Theorem bnj976
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 bnj976.4 . 2  |-  ( ch'  <->  [. G  /  f ]. ch )
2 sbcco 3292 . 2  |-  ( [. G  /  h ]. [. h  /  f ]. ch  <->  [. G  /  f ]. ch )
3 bnj976.5 . . 3  |-  G  e. 
_V
4 bnj252 29520 . . . . . 6  |-  ( ( N  e.  D  /\  f  Fn  N  /\  ph 
/\  ps )  <->  ( N  e.  D  /\  (
f  Fn  N  /\  ph 
/\  ps ) ) )
54sbcbii 3325 . . . . 5  |-  ( [. h  /  f ]. ( N  e.  D  /\  f  Fn  N  /\  ph 
/\  ps )  <->  [. h  / 
f ]. ( N  e.  D  /\  ( f  Fn  N  /\  ph  /\ 
ps ) ) )
6 bnj976.1 . . . . . 6  |-  ( ch  <->  ( N  e.  D  /\  f  Fn  N  /\  ph 
/\  ps ) )
76sbcbii 3325 . . . . 5  |-  ( [. h  /  f ]. ch  <->  [. h  /  f ]. ( N  e.  D  /\  f  Fn  N  /\  ph  /\  ps )
)
8 vex 3050 . . . . . . . 8  |-  h  e. 
_V
98bnj525 29559 . . . . . . 7  |-  ( [. h  /  f ]. N  e.  D  <->  N  e.  D
)
10 sbc3an 3327 . . . . . . . 8  |-  ( [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps )  <->  ( [. h  /  f ]. f  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
11 bnj62 29538 . . . . . . . . 9  |-  ( [. h  /  f ]. f  Fn  N  <->  h  Fn  N
)
12113anbi1i 1200 . . . . . . . 8  |-  ( (
[. h  /  f ]. f  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
1310, 12bitri 253 . . . . . . 7  |-  ( [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps )  <->  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
149, 13anbi12i 704 . . . . . 6  |-  ( (
[. h  /  f ]. N  e.  D  /\  [. h  /  f ]. ( f  Fn  N  /\  ph  /\  ps )
)  <->  ( N  e.  D  /\  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) ) )
15 sbcan 3312 . . . . . 6  |-  ( [. h  /  f ]. ( N  e.  D  /\  ( f  Fn  N  /\  ph  /\  ps )
)  <->  ( [. h  /  f ]. N  e.  D  /\  [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps ) ) )
16 bnj252 29520 . . . . . 6  |-  ( ( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph 
/\  [. h  /  f ]. ps )  <->  ( N  e.  D  /\  (
h  Fn  N  /\  [. h  /  f ]. ph 
/\  [. h  /  f ]. ps ) ) )
1714, 15, 163bitr4ri 282 . . . . 5  |-  ( ( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph 
/\  [. h  /  f ]. ps )  <->  [. h  / 
f ]. ( N  e.  D  /\  ( f  Fn  N  /\  ph  /\ 
ps ) ) )
185, 7, 173bitr4i 281 . . . 4  |-  ( [. h  /  f ]. ch  <->  ( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph 
/\  [. h  /  f ]. ps ) )
19 fneq1 5669 . . . . . . 7  |-  ( h  =  G  ->  (
h  Fn  N  <->  G  Fn  N ) )
20 sbceq1a 3280 . . . . . . . 8  |-  ( h  =  G  ->  ( [. h  /  f ]. ph  <->  [. G  /  h ]. [. h  /  f ]. ph ) )
21 bnj976.2 . . . . . . . . 9  |-  ( ph'  <->  [. G  /  f ]. ph )
22 sbcco 3292 . . . . . . . . 9  |-  ( [. G  /  h ]. [. h  /  f ]. ph  <->  [. G  / 
f ]. ph )
2321, 22bitr4i 256 . . . . . . . 8  |-  ( ph'  <->  [. G  /  h ]. [. h  /  f ]. ph )
2420, 23syl6bbr 267 . . . . . . 7  |-  ( h  =  G  ->  ( [. h  /  f ]. ph  <->  ph' ) )
25 sbceq1a 3280 . . . . . . . 8  |-  ( h  =  G  ->  ( [. h  /  f ]. ps  <->  [. G  /  h ]. [. h  /  f ]. ps ) )
26 bnj976.3 . . . . . . . . 9  |-  ( ps'  <->  [. G  /  f ]. ps )
27 sbcco 3292 . . . . . . . . 9  |-  ( [. G  /  h ]. [. h  /  f ]. ps  <->  [. G  /  f ]. ps )
2826, 27bitr4i 256 . . . . . . . 8  |-  ( ps'  <->  [. G  /  h ]. [. h  /  f ]. ps )
2925, 28syl6bbr 267 . . . . . . 7  |-  ( h  =  G  ->  ( [. h  /  f ]. ps  <->  ps' ) )
3019, 24, 293anbi123d 1341 . . . . . 6  |-  ( h  =  G  ->  (
( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( G  Fn  N  /\  ph'  /\  ps' ) ) )
3130anbi2d 711 . . . . 5  |-  ( h  =  G  ->  (
( N  e.  D  /\  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )  <->  ( N  e.  D  /\  ( G  Fn  N  /\  ph' 
/\  ps' ) ) ) )
32 bnj252 29520 . . . . 5  |-  ( ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' )  <->  ( N  e.  D  /\  ( G  Fn  N  /\  ph' 
/\  ps' ) ) )
3331, 16, 323bitr4g 292 . . . 4  |-  ( h  =  G  ->  (
( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( N  e.  D  /\  G  Fn  N  /\  ph'  /\  ps' ) ) )
3418, 33syl5bb 261 . . 3  |-  ( h  =  G  ->  ( [. h  /  f ]. ch  <->  ( N  e.  D  /\  G  Fn  N  /\  ph'  /\  ps' ) ) )
353, 34sbcie 3304 . 2  |-  ( [. G  /  h ]. [. h  /  f ]. ch  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
361, 2, 353bitr2i 277 1  |-  ( ch'  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   _Vcvv 3047   [.wsbc 3269    Fn wfn 5580    /\ w-bnj17 29503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-fun 5587  df-fn 5588  df-bnj17 29504
This theorem is referenced by:  bnj910  29771  bnj999  29780  bnj907  29788
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