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Theorem bnj62 29062
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj62  |-  ( [. z  /  x ]. x  Fn  A  <->  z  Fn  A
)
Distinct variable group:    x, A
Allowed substitution hint:    A( z)

Proof of Theorem bnj62
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . 4  |-  y  e. 
_V
2 fneq1 5349 . . . 4  |-  ( x  =  y  ->  (
x  Fn  A  <->  y  Fn  A ) )
31, 2sbcie 3038 . . 3  |-  ( [. y  /  x ]. x  Fn  A  <->  y  Fn  A
)
43sbcbii 3059 . 2  |-  ( [. z  /  y ]. [. y  /  x ]. x  Fn  A  <->  [. z  /  y ]. y  Fn  A
)
5 sbcco 3026 . 2  |-  ( [. z  /  y ]. [. y  /  x ]. x  Fn  A  <->  [. z  /  x ]. x  Fn  A
)
6 vex 2804 . . 3  |-  z  e. 
_V
7 fneq1 5349 . . 3  |-  ( y  =  z  ->  (
y  Fn  A  <->  z  Fn  A ) )
86, 7sbcie 3038 . 2  |-  ( [. z  /  y ]. y  Fn  A  <->  z  Fn  A
)
94, 5, 83bitr3i 266 1  |-  ( [. z  /  x ]. x  Fn  A  <->  z  Fn  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [.wsbc 3004    Fn wfn 5266
This theorem is referenced by:  bnj156  29072  bnj976  29125  bnj581  29256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274
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