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Theorem bnj90 29024
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 5527 . . 3  |-  ( x  =  y  ->  (
z  Fn  x  <->  z  Fn  y ) )
3 fneq2 5527 . . 3  |-  ( y  =  Y  ->  (
z  Fn  y  <->  z  Fn  Y ) )
42, 3sbcie2g 3186 . 2  |-  ( Y  e.  _V  ->  ( [. Y  /  x ]. z  Fn  x  <->  z  Fn  Y ) )
51, 4ax-mp 8 1  |-  ( [. Y  /  x ]. z  Fn  x  <->  z  Fn  Y
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   _Vcvv 2948   [.wsbc 3153    Fn wfn 5441
This theorem is referenced by:  bnj121  29178  bnj130  29182  bnj207  29189
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 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-fn 5449
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