MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralunsn Structured version   Unicode version

Theorem ralunsn 4168
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 3616 . 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 3996 . . 3  |-  ( B  e.  C  ->  ( A. x  e.  { B } ph  <->  ps ) )
43anbi2d 701 . 2  |-  ( B  e.  C  ->  (
( A. x  e.  A  ph  /\  A. x  e.  { B } ph )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
51, 4syl5bb 257 1  |-  ( B  e.  C  ->  ( A. x  e.  ( A  u.  { B } ) ph  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746    u. cun 3404   {csn 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ral 2751  df-v 3053  df-sbc 3270  df-un 3411  df-sn 3962
This theorem is referenced by:  2ralunsn  4169  symgextfo  16587  gsmsymgrfixlem1  16592  gsmsymgreqlem2  16596  symgfixf1  16602  cply1coe0bi  18478  scmatf1  19141  mdetunilem9  19230  m2cpminvid2lem  19363  clwlkisclwwlklem2a1  24925  clwlkf1clwwlklem  24995  disjunsn  27617
  Copyright terms: Public domain W3C validator