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Theorem bnj156 29032
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj156.1  |-  ( ze0  <->  (
f  Fn  1o  /\  ph' 
/\  ps' ) )
bnj156.2  |-  ( ze1  <->  [. g  /  f ]. ze0 )
bnj156.3  |-  ( ph1  <->  [. g  /  f ]. ph' )
bnj156.4  |-  ( ps1  <->  [. g  /  f ]. ps' )
Assertion
Ref Expression
bnj156  |-  ( ze1  <->  (
g  Fn  1o  /\  ph1 
/\  ps1 ) )

Proof of Theorem bnj156
StepHypRef Expression
1 bnj156.2 . 2  |-  ( ze1  <->  [. g  /  f ]. ze0 )
2 bnj156.1 . . . 4  |-  ( ze0  <->  (
f  Fn  1o  /\  ph' 
/\  ps' ) )
32sbcbii 3208 . . 3  |-  ( [. g  /  f ]. ze0  <->  [. g  / 
f ]. ( f  Fn  1o  /\  ph'  /\  ps' ) )
4 vex 2951 . . . . 5  |-  g  e. 
_V
5 sbc3ang 3211 . . . . 5  |-  ( g  e.  _V  ->  ( [. g  /  f ]. ( f  Fn  1o  /\  ph'  /\  ps' )  <->  ( [. g  /  f ]. f  Fn  1o  /\  [. g  /  f ]. ph'  /\  [. g  /  f ]. ps' ) ) )
64, 5ax-mp 8 . . . 4  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( [. g  /  f ]. f  Fn  1o  /\  [. g  /  f ]. ph'  /\  [. g  /  f ]. ps' ) )
7 bnj62 29022 . . . . 5  |-  ( [. g  /  f ]. f  Fn  1o  <->  g  Fn  1o )
8 bnj156.3 . . . . . 6  |-  ( ph1  <->  [. g  /  f ]. ph' )
98bicomi 194 . . . . 5  |-  ( [. g  /  f ]. ph'  <->  ph1 )
10 bnj156.4 . . . . . 6  |-  ( ps1  <->  [. g  /  f ]. ps' )
1110bicomi 194 . . . . 5  |-  ( [. g  /  f ]. ps'  <->  ps1 )
127, 9, 113anbi123i 1142 . . . 4  |-  ( (
[. g  /  f ]. f  Fn  1o  /\ 
[. g  /  f ]. ph'  /\  [. g  /  f ]. ps' )  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
136, 12bitri 241 . . 3  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
143, 13bitri 241 . 2  |-  ( [. g  /  f ]. ze0  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
151, 14bitri 241 1  |-  ( ze1  <->  (
g  Fn  1o  /\  ph1 
/\  ps1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ w3a 936    e. wcel 1725   _Vcvv 2948   [.wsbc 3153    Fn wfn 5441   1oc1o 6709
This theorem is referenced by:  bnj153  29188
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448  df-fn 5449
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