Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj90 Structured version   Unicode version

Theorem bnj90 33483
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 5656 . . 3  |-  ( x  =  y  ->  (
z  Fn  x  <->  z  Fn  y ) )
3 fneq2 5656 . . 3  |-  ( y  =  Y  ->  (
z  Fn  y  <->  z  Fn  Y ) )
42, 3sbcie2g 3345 . 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 1802   _Vcvv 3093   [.wsbc 3311    Fn wfn 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-v 3095  df-sbc 3312  df-fn 5577
This theorem is referenced by:  bnj121  33635  bnj130  33639  bnj207  33646
  Copyright terms: Public domain W3C validator