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Theorem bnj90 33072
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj90.1  |-  Y  e. 
_V
Assertion
Ref Expression
bnj90  |-  ( [. Y  /  x ]. z  Fn  x  <->  z  Fn  Y
)
Distinct variable group:    x, z
Allowed substitution hints:    Y( x, z)

Proof of Theorem bnj90
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 bnj90.1 . 2  |-  Y  e. 
_V
2 fneq2 5670 . . 3  |-  ( x  =  y  ->  (
z  Fn  x  <->  z  Fn  y ) )
3 fneq2 5670 . . 3  |-  ( y  =  Y  ->  (
z  Fn  y  <->  z  Fn  Y ) )
42, 3sbcie2g 3365 . 2  |-  ( Y  e.  _V  ->  ( [. Y  /  x ]. z  Fn  x  <->  z  Fn  Y ) )
51, 4ax-mp 5 1  |-  ( [. Y  /  x ]. z  Fn  x  <->  z  Fn  Y
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1767   _Vcvv 3113   [.wsbc 3331    Fn wfn 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332  df-fn 5591
This theorem is referenced by:  bnj121  33224  bnj130  33228  bnj207  33235
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