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Theorem abeq2f 23965
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 2567 . . 3  |-  ( y  e.  A  ->  A. x  y  e.  A )
3 hbab1 2427 . . 3  |-  ( y  e.  { x  | 
ph }  ->  A. x  y  e.  { x  |  ph } )
42, 3cleqh 2535 . 2  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  x  e.  { x  | 
ph } ) )
5 abid 2426 . . . 4  |-  ( x  e.  { x  | 
ph }  <->  ph )
65bibi2i 306 . . 3  |-  ( ( x  e.  A  <->  x  e.  { x  |  ph }
)  <->  ( x  e.  A  <->  ph ) )
76albii 1576 . 2  |-  ( A. x ( x  e.  A  <->  x  e.  { x  |  ph } )  <->  A. x
( x  e.  A  <->  ph ) )
84, 7bitri 242 1  |-  ( A  =  { x  | 
ph }  <->  A. x
( x  e.  A  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   {cab 2424   F/_wnfc 2561
This theorem is referenced by:  rabid2f  23972  mptfnf  24078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563
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