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Theorem ralunsn 3815
Description: Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypothesis
Ref Expression
ralunsn.1  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ralunsn  |-  ( B  e.  C  ->  ( A. x  e.  ( A  u.  { B } ) ph  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Distinct variable groups:    x, B    ps, x
Allowed substitution hints:    ph( x)    A( x)    C( x)

Proof of Theorem ralunsn
StepHypRef Expression
1 ralunb 3356 . 2  |-  ( A. x  e.  ( A  u.  { B } )
ph 
<->  ( A. x  e.  A  ph  /\  A. x  e.  { B } ph ) )
2 ralunsn.1 . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
32ralsng 3672 . . 3  |-  ( B  e.  C  ->  ( A. x  e.  { B } ph  <->  ps ) )
43anbi2d 684 . 2  |-  ( B  e.  C  ->  (
( A. x  e.  A  ph  /\  A. x  e.  { B } ph )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
51, 4syl5bb 248 1  |-  ( B  e.  C  ->  ( A. x  e.  ( A  u.  { B } ) ph  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150   {csn 3640
This theorem is referenced by:  2ralunsn  3816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646
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