Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abeq2f Unicode version

Theorem abeq2f 23948
Description: Equality of a class variable and a class abstraction. In this version, the fact that  x is a non-free variable in  A is explicitely stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)
Hypothesis
Ref Expression
abeq2f.0  |-  F/_ x A
Assertion
Ref Expression
abeq2f  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )

Proof of Theorem abeq2f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 abeq2f.0 . . . 4  |-  F/_ x A
21nfcrii 2564 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
3 hbab1 2424 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
42, 3cleqh 2532 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  x  e.  { x  | 
ph } ) )
5 abid 2423 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
65bibi2i 305 . . 3  |-  ( ( x  e.  A  <->  x  e.  { x  |  ph }
)  <->  ( x  e.  A  <->  ph ) )
76albii 1575 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ph } )  <->  A. x
( x  e.  A  <->  ph ) )
84, 7bitri 241 1  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   {cab 2421   F/_wnfc 2558
This theorem is referenced by:  rabid2f  23955  mptfnf  24061
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
  Copyright terms: Public domain W3C validator