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Theorem bnj156 32740
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 3386 . . 3  |-  ( [. g  /  f ]. ze0  <->  [. g  / 
f ]. ( f  Fn  1o  /\  ph'  /\  ps' ) )
4 sbc3an 3389 . . . 4  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( [. g  /  f ]. f  Fn  1o  /\  [. g  /  f ]. ph'  /\  [. g  /  f ]. ps' ) )
5 bnj62 32730 . . . . 5  |-  ( [. g  /  f ]. f  Fn  1o  <->  g  Fn  1o )
6 bnj156.3 . . . . . 6  |-  ( ph1  <->  [. g  /  f ]. ph' )
76bicomi 202 . . . . 5  |-  ( [. g  /  f ]. ph'  <->  ph1 )
8 bnj156.4 . . . . . 6  |-  ( ps1  <->  [. g  /  f ]. ps' )
98bicomi 202 . . . . 5  |-  ( [. g  /  f ]. ps'  <->  ps1 )
105, 7, 93anbi123i 1180 . . . 4  |-  ( (
[. g  /  f ]. f  Fn  1o  /\ 
[. g  /  f ]. ph'  /\  [. g  /  f ]. ps' )  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
114, 10bitri 249 . . 3  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  ph' 
/\  ps' )  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
123, 11bitri 249 . 2  |-  ( [. g  /  f ]. ze0  <->  ( g  Fn  1o  /\  ph1  /\  ps1 )
)
131, 12bitri 249 1  |-  ( ze1  <->  (
g  Fn  1o  /\  ph1 
/\  ps1 ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 968   [.wsbc 3326    Fn wfn 5576   1oc1o 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-fun 5583  df-fn 5584
This theorem is referenced by:  bnj153  32894
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