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Theorem bnj62 29085
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 2959 . . . 4  |-  y  e. 
_V
2 fneq1 5534 . . . 4  |-  ( x  =  y  ->  (
x  Fn  A  <->  y  Fn  A ) )
31, 2sbcie 3195 . . 3  |-  ( [. y  /  x ]. x  Fn  A  <->  y  Fn  A
)
43sbcbii 3216 . 2  |-  ( [. z  /  y ]. [. y  /  x ]. x  Fn  A  <->  [. z  /  y ]. y  Fn  A
)
5 sbcco 3183 . 2  |-  ( [. z  /  y ]. [. y  /  x ]. x  Fn  A  <->  [. z  /  x ]. x  Fn  A
)
6 vex 2959 . . 3  |-  z  e. 
_V
7 fneq1 5534 . . 3  |-  ( y  =  z  ->  (
y  Fn  A  <->  z  Fn  A ) )
86, 7sbcie 3195 . 2  |-  ( [. z  /  y ]. y  Fn  A  <->  z  Fn  A
)
94, 5, 83bitr3i 267 1  |-  ( [. z  /  x ]. x  Fn  A  <->  z  Fn  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [.wsbc 3161    Fn wfn 5449
This theorem is referenced by:  bnj156  29095  bnj976  29148  bnj581  29279
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-fun 5456  df-fn 5457
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