Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj976 Structured version   Unicode version

Theorem bnj976 29148
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 3183 . 2  |-  ( [. G  /  h ]. [. h  /  f ]. ch  <->  [. G  /  f ]. ch )
3 bnj976.5 . . 3  |-  G  e. 
_V
4 bnj252 29067 . . . . . 6  |-  ( ( N  e.  D  /\  f  Fn  N  /\  ph 
/\  ps )  <->  ( N  e.  D  /\  (
f  Fn  N  /\  ph 
/\  ps ) ) )
54sbcbii 3216 . . . . 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 3216 . . . . 5  |-  ( [. h  /  f ]. ch  <->  [. h  /  f ]. ( N  e.  D  /\  f  Fn  N  /\  ph  /\  ps )
)
8 vex 2959 . . . . . . . 8  |-  h  e. 
_V
98bnj525 29106 . . . . . . 7  |-  ( [. h  /  f ]. N  e.  D  <->  N  e.  D
)
10 sbc3ang 3219 . . . . . . . . 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 29085 . . . . . . . . 9  |-  ( [. h  /  f ]. f  Fn  N  <->  h  Fn  N
)
13123anbi1i 1144 . . . . . . . 8  |-  ( (
[. h  /  f ]. f  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
1411, 13bitri 241 . . . . . . 7  |-  ( [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps )  <->  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
159, 14anbi12i 679 . . . . . 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 3203 . . . . . 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 29067 . . . . . 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 270 . . . . 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 269 . . . 4  |-  ( [. h  /  f ]. ch  <->  ( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph 
/\  [. h  /  f ]. ps ) )
20 fneq1 5534 . . . . . . 7  |-  ( h  =  G  ->  (
h  Fn  N  <->  G  Fn  N ) )
21 sbceq1a 3171 . . . . . . . 8  |-  ( h  =  G  ->  ( [. h  /  f ]. ph  <->  [. G  /  h ]. [. h  /  f ]. ph ) )
22 bnj976.2 . . . . . . . . 9  |-  ( ph'  <->  [. G  /  f ]. ph )
23 sbcco 3183 . . . . . . . . 9  |-  ( [. G  /  h ]. [. h  /  f ]. ph  <->  [. G  / 
f ]. ph )
2422, 23bitr4i 244 . . . . . . . 8  |-  ( ph'  <->  [. G  /  h ]. [. h  /  f ]. ph )
2521, 24syl6bbr 255 . . . . . . 7  |-  ( h  =  G  ->  ( [. h  /  f ]. ph  <->  ph' ) )
26 sbceq1a 3171 . . . . . . . 8  |-  ( h  =  G  ->  ( [. h  /  f ]. ps  <->  [. G  /  h ]. [. h  /  f ]. ps ) )
27 bnj976.3 . . . . . . . . 9  |-  ( ps'  <->  [. G  /  f ]. ps )
28 sbcco 3183 . . . . . . . . 9  |-  ( [. G  /  h ]. [. h  /  f ]. ps  <->  [. G  /  f ]. ps )
2927, 28bitr4i 244 . . . . . . . 8  |-  ( ps'  <->  [. G  /  h ]. [. h  /  f ]. ps )
3026, 29syl6bbr 255 . . . . . . 7  |-  ( h  =  G  ->  ( [. h  /  f ]. ps  <->  ps' ) )
3120, 25, 303anbi123d 1254 . . . . . 6  |-  ( h  =  G  ->  (
( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( G  Fn  N  /\  ph'  /\  ps' ) ) )
3231anbi2d 685 . . . . 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 29067 . . . . 5  |-  ( ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' )  <->  ( N  e.  D  /\  ( G  Fn  N  /\  ph' 
/\  ps' ) ) )
3432, 17, 333bitr4g 280 . . . 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 249 . . 3  |-  ( h  =  G  ->  ( [. h  /  f ]. ch  <->  ( N  e.  D  /\  G  Fn  N  /\  ph'  /\  ps' ) ) )
363, 35sbcie 3195 . 2  |-  ( [. G  /  h ]. [. h  /  f ]. ch  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
371, 2, 363bitr2i 265 1  |-  ( ch'  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956   [.wsbc 3161    Fn wfn 5449    /\ w-bnj17 29050
This theorem is referenced by:  bnj910  29319  bnj999  29328  bnj907  29336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-fun 5456  df-fn 5457  df-bnj17 29051
  Copyright terms: Public domain W3C validator