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Theorem bnj976 29125
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 3026 . 2  |-  ( [. G  /  h ]. [. h  /  f ]. ch  <->  [. G  /  f ]. ch )
3 bnj976.5 . . 3  |-  G  e. 
_V
4 bnj252 29044 . . . . . 6  |-  ( ( N  e.  D  /\  f  Fn  N  /\  ph 
/\  ps )  <->  ( N  e.  D  /\  (
f  Fn  N  /\  ph 
/\  ps ) ) )
54sbcbii 3059 . . . . 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 3059 . . . . 5  |-  ( [. h  /  f ]. ch  <->  [. h  /  f ]. ( N  e.  D  /\  f  Fn  N  /\  ph  /\  ps )
)
8 vex 2804 . . . . . . . 8  |-  h  e. 
_V
98bnj525 29083 . . . . . . 7  |-  ( [. h  /  f ]. N  e.  D  <->  N  e.  D
)
10 sbc3ang 3062 . . . . . . . . 9  |-  ( h  e.  _V  ->  ( [. h  /  f ]. ( f  Fn  N  /\  ph  /\  ps )  <->  (
[. h  /  f ]. f  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) ) )
118, 10ax-mp 8 . . . . . . . 8  |-  ( [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps )  <->  ( [. h  /  f ]. f  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
12 bnj62 29062 . . . . . . . . 9  |-  ( [. h  /  f ]. f  Fn  N  <->  h  Fn  N
)
13123anbi1i 1142 . . . . . . . 8  |-  ( (
[. h  /  f ]. f  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
1411, 13bitri 240 . . . . . . 7  |-  ( [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps )  <->  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
159, 14anbi12i 678 . . . . . 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 ) ) )
16 sbcan 3046 . . . . . 6  |-  ( [. h  /  f ]. ( N  e.  D  /\  ( f  Fn  N  /\  ph  /\  ps )
)  <->  ( [. h  /  f ]. N  e.  D  /\  [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps ) ) )
17 bnj252 29044 . . . . . 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 ) ) )
1815, 16, 173bitr4ri 269 . . . . 5  |-  ( ( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph 
/\  [. h  /  f ]. ps )  <->  [. h  / 
f ]. ( N  e.  D  /\  ( f  Fn  N  /\  ph  /\ 
ps ) ) )
195, 7, 183bitr4i 268 . . . 4  |-  ( [. h  /  f ]. ch  <->  ( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph 
/\  [. h  /  f ]. ps ) )
20 fneq1 5349 . . . . . . 7  |-  ( h  =  G  ->  (
h  Fn  N  <->  G  Fn  N ) )
21 sbceq1a 3014 . . . . . . . 8  |-  ( h  =  G  ->  ( [. h  /  f ]. ph  <->  [. G  /  h ]. [. h  /  f ]. ph ) )
22 bnj976.2 . . . . . . . . 9  |-  ( ph'  <->  [. G  /  f ]. ph )
23 sbcco 3026 . . . . . . . . 9  |-  ( [. G  /  h ]. [. h  /  f ]. ph  <->  [. G  / 
f ]. ph )
2422, 23bitr4i 243 . . . . . . . 8  |-  ( ph'  <->  [. G  /  h ]. [. h  /  f ]. ph )
2521, 24syl6bbr 254 . . . . . . 7  |-  ( h  =  G  ->  ( [. h  /  f ]. ph  <->  ph' ) )
26 sbceq1a 3014 . . . . . . . 8  |-  ( h  =  G  ->  ( [. h  /  f ]. ps  <->  [. G  /  h ]. [. h  /  f ]. ps ) )
27 bnj976.3 . . . . . . . . 9  |-  ( ps'  <->  [. G  /  f ]. ps )
28 sbcco 3026 . . . . . . . . 9  |-  ( [. G  /  h ]. [. h  /  f ]. ps  <->  [. G  /  f ]. ps )
2927, 28bitr4i 243 . . . . . . . 8  |-  ( ps'  <->  [. G  /  h ]. [. h  /  f ]. ps )
3026, 29syl6bbr 254 . . . . . . 7  |-  ( h  =  G  ->  ( [. h  /  f ]. ps  <->  ps' ) )
3120, 25, 303anbi123d 1252 . . . . . 6  |-  ( h  =  G  ->  (
( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( G  Fn  N  /\  ph'  /\  ps' ) ) )
3231anbi2d 684 . . . . 5  |-  ( h  =  G  ->  (
( N  e.  D  /\  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )  <->  ( N  e.  D  /\  ( G  Fn  N  /\  ph' 
/\  ps' ) ) ) )
33 bnj252 29044 . . . . 5  |-  ( ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' )  <->  ( N  e.  D  /\  ( G  Fn  N  /\  ph' 
/\  ps' ) ) )
3432, 17, 333bitr4g 279 . . . 4  |-  ( h  =  G  ->  (
( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( N  e.  D  /\  G  Fn  N  /\  ph'  /\  ps' ) ) )
3519, 34syl5bb 248 . . 3  |-  ( h  =  G  ->  ( [. h  /  f ]. ch  <->  ( N  e.  D  /\  G  Fn  N  /\  ph'  /\  ps' ) ) )
363, 35sbcie 3038 . 2  |-  ( [. G  /  h ]. [. h  /  f ]. ch  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
371, 2, 363bitr2i 264 1  |-  ( ch'  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004    Fn wfn 5266    /\ w-bnj17 29027
This theorem is referenced by:  bnj910  29296  bnj999  29305  bnj907  29313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274  df-bnj17 29028
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