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Theorem bnj62 28746
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 2791 . . . 4  |-  y  e. 
_V
2 fneq1 5333 . . . 4  |-  ( x  =  y  ->  (
x  Fn  A  <->  y  Fn  A ) )
31, 2sbcie 3025 . . 3  |-  ( [. y  /  x ]. x  Fn  A  <->  y  Fn  A
)
43sbcbii 3046 . 2  |-  ( [. z  /  y ]. [. y  /  x ]. x  Fn  A  <->  [. z  /  y ]. y  Fn  A
)
5 sbcco 3013 . 2  |-  ( [. z  /  y ]. [. y  /  x ]. x  Fn  A  <->  [. z  /  x ]. x  Fn  A
)
6 vex 2791 . . 3  |-  z  e. 
_V
7 fneq1 5333 . . 3  |-  ( y  =  z  ->  (
y  Fn  A  <->  z  Fn  A ) )
86, 7sbcie 3025 . 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 2991    Fn wfn 5250
This theorem is referenced by:  bnj156  28756  bnj976  28809  bnj581  28940
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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