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Theorem bnj976 31769
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 3208 . 2  |-  ( [. G  /  h ]. [. h  /  f ]. ch  <->  [. G  /  f ]. ch )
3 bnj976.5 . . 3  |-  G  e. 
_V
4 bnj252 31689 . . . . . 6  |-  ( ( N  e.  D  /\  f  Fn  N  /\  ph 
/\  ps )  <->  ( N  e.  D  /\  (
f  Fn  N  /\  ph 
/\  ps ) ) )
54sbcbii 3245 . . . . 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 3245 . . . . 5  |-  ( [. h  /  f ]. ch  <->  [. h  /  f ]. ( N  e.  D  /\  f  Fn  N  /\  ph  /\  ps )
)
8 vex 2974 . . . . . . . 8  |-  h  e. 
_V
98bnj525 31728 . . . . . . 7  |-  ( [. h  /  f ]. N  e.  D  <->  N  e.  D
)
10 sbc3an 3248 . . . . . . . 8  |-  ( [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps )  <->  ( [. h  /  f ]. f  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
11 bnj62 31707 . . . . . . . . 9  |-  ( [. h  /  f ]. f  Fn  N  <->  h  Fn  N
)
12113anbi1i 1178 . . . . . . . 8  |-  ( (
[. h  /  f ]. f  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
1310, 12bitri 249 . . . . . . 7  |-  ( [. h  /  f ]. (
f  Fn  N  /\  ph 
/\  ps )  <->  ( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) )
149, 13anbi12i 697 . . . . . 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 3228 . . . . . 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 31689 . . . . . 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 278 . . . . 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 277 . . . 4  |-  ( [. h  /  f ]. ch  <->  ( N  e.  D  /\  h  Fn  N  /\  [. h  /  f ]. ph 
/\  [. h  /  f ]. ps ) )
19 fneq1 5498 . . . . . . 7  |-  ( h  =  G  ->  (
h  Fn  N  <->  G  Fn  N ) )
20 sbceq1a 3196 . . . . . . . 8  |-  ( h  =  G  ->  ( [. h  /  f ]. ph  <->  [. G  /  h ]. [. h  /  f ]. ph ) )
21 bnj976.2 . . . . . . . . 9  |-  ( ph'  <->  [. G  /  f ]. ph )
22 sbcco 3208 . . . . . . . . 9  |-  ( [. G  /  h ]. [. h  /  f ]. ph  <->  [. G  / 
f ]. ph )
2321, 22bitr4i 252 . . . . . . . 8  |-  ( ph'  <->  [. G  /  h ]. [. h  /  f ]. ph )
2420, 23syl6bbr 263 . . . . . . 7  |-  ( h  =  G  ->  ( [. h  /  f ]. ph  <->  ph' ) )
25 sbceq1a 3196 . . . . . . . 8  |-  ( h  =  G  ->  ( [. h  /  f ]. ps  <->  [. G  /  h ]. [. h  /  f ]. ps ) )
26 bnj976.3 . . . . . . . . 9  |-  ( ps'  <->  [. G  /  f ]. ps )
27 sbcco 3208 . . . . . . . . 9  |-  ( [. G  /  h ]. [. h  /  f ]. ps  <->  [. G  /  f ]. ps )
2826, 27bitr4i 252 . . . . . . . 8  |-  ( ps'  <->  [. G  /  h ]. [. h  /  f ]. ps )
2925, 28syl6bbr 263 . . . . . . 7  |-  ( h  =  G  ->  ( [. h  /  f ]. ps  <->  ps' ) )
3019, 24, 293anbi123d 1289 . . . . . 6  |-  ( h  =  G  ->  (
( h  Fn  N  /\  [. h  /  f ]. ph  /\  [. h  /  f ]. ps ) 
<->  ( G  Fn  N  /\  ph'  /\  ps' ) ) )
3130anbi2d 703 . . . . 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 31689 . . . . 5  |-  ( ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' )  <->  ( N  e.  D  /\  ( G  Fn  N  /\  ph' 
/\  ps' ) ) )
3331, 16, 323bitr4g 288 . . . 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 257 . . 3  |-  ( h  =  G  ->  ( [. h  /  f ]. ch  <->  ( N  e.  D  /\  G  Fn  N  /\  ph'  /\  ps' ) ) )
353, 34sbcie 3220 . 2  |-  ( [. G  /  h ]. [. h  /  f ]. ch  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
361, 2, 353bitr2i 273 1  |-  ( ch'  <->  ( N  e.  D  /\  G  Fn  N  /\  ph' 
/\  ps' ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2971   [.wsbc 3185    Fn wfn 5412    /\ w-bnj17 31672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-fun 5419  df-fn 5420  df-bnj17 31673
This theorem is referenced by:  bnj910  31939  bnj999  31948  bnj907  31956
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