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Theorem ralunsn 3690
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 3223 . 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 3530 . . 3  |-  ( B  e.  C  ->  ( A. x  e.  { B } ph  <->  ps ) )
43anbi2d 457 . 2  |-  ( B  e.  C  ->  (
( A. x  e.  A  ph  /\  A. x  e.  { B } ph )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
51, 4syl5bb 191 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    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2390    u. cun 3035   {csn 3493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-sbc 2879  df-un 3041  df-sn 3499
This theorem is referenced by:  2ralunsn  3691
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