ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralunsn Unicode version

Theorem ralunsn 3597
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 3154 . 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 3441 . . 3  |-  ( B  e.  C  ->  ( A. x  e.  { B } ph  <->  ps ) )
43anbi2d 452 . 2  |-  ( B  e.  C  ->  (
( A. x  e.  A  ph  /\  A. x  e.  { B } ph )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
51, 4syl5bb 190 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349    u. cun 2972   {csn 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-sbc 2817  df-un 2978  df-sn 3412
This theorem is referenced by:  2ralunsn  3598
  Copyright terms: Public domain W3C validator