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Theorem bnj156 28756
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 3046 . . 3  |-  ( [. g  /  f ]. ze0  <->  [. g  / 
f ]. ( f  Fn  1o  /\  ph'  /\  ps' ) )
4 vex 2791 . . . . 5  |-  g  e. 
_V
5 sbc3ang 3049 . . . . 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 28746 . . . . 5  |-  ( [. g  /  f ]. f  Fn  1o  <->  g  Fn  1o )
8 bnj156.3 . . . . . 6  |-  ( ph1  <->  [. g  /  f ]. ph' )
98bicomi 193 . . . . 5  |-  ( [. g  /  f ]. ph'  <->  ph1 )
10 bnj156.4 . . . . . 6  |-  ( ps1  <->  [. g  /  f ]. ps' )
1110bicomi 193 . . . . 5  |-  ( [. g  /  f ]. ps'  <->  ps1 )
127, 9, 113anbi123i 1140 . . . 4  |-  ( (
[. g  /  f ]. f  Fn  1o  /\ 
[. g  /  f ]. ph'  /\  [. g  /  f ]. ps' )  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
136, 12bitri 240 . . 3  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
143, 13bitri 240 . 2  |-  ( [. g  /  f ]. ze0  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
151, 14bitri 240 1  |-  ( ze1  <->  (
g  Fn  1o  /\  ph1 
/\  ps1 ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ w3a 934    e. wcel 1684   _Vcvv 2788   [.wsbc 2991    Fn wfn 5250   1oc1o 6472
This theorem is referenced by:  bnj153  28912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
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