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Theorem abbi1dvOLD 2556
Description: Obsolete proof of abbi1dv 2555 as of 16-Nov-2019. (Contributed by NM, 9-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
abbi1dv.1  |-  ( ph  ->  ( ps  <->  x  e.  A ) )
Assertion
Ref Expression
abbi1dvOLD  |-  ( ph  ->  { x  |  ps }  =  A )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem abbi1dvOLD
StepHypRef Expression
1 abbi1dv.1 . . 3  |-  ( ph  ->  ( ps  <->  x  e.  A ) )
21alrimiv 1767 . 2  |-  ( ph  ->  A. x ( ps  <->  x  e.  A ) )
3 abeq1 2542 . 2  |-  ( { x  |  ps }  =  A  <->  A. x ( ps  <->  x  e.  A ) )
42, 3sylibr 215 1  |-  ( ph  ->  { x  |  ps }  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437    e. wcel 1872   {cab 2407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417
This theorem is referenced by: (None)
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