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

Theorem ralprg 4076
Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ralprg.2  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
ralprg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
Distinct variable groups:    x, A    x, B    ps, x    ch, x
Allowed substitution hints:    ph( x)    V( x)    W( x)

Proof of Theorem ralprg
StepHypRef Expression
1 df-pr 4030 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
21raleqi 3062 . . 3  |-  ( A. x  e.  { A ,  B } ph  <->  A. x  e.  ( { A }  u.  { B } )
ph )
3 ralunb 3685 . . 3  |-  ( A. x  e.  ( { A }  u.  { B } ) ph  <->  ( A. x  e.  { A } ph  /\  A. x  e.  { B } ph ) )
42, 3bitri 249 . 2  |-  ( A. x  e.  { A ,  B } ph  <->  ( A. x  e.  { A } ph  /\  A. x  e.  { B } ph ) )
5 ralprg.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
65ralsng 4062 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  ps ) )
7 ralprg.2 . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
87ralsng 4062 . . 3  |-  ( B  e.  W  ->  ( A. x  e.  { B } ph  <->  ch ) )
96, 8bi2anan9 871 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A. x  e.  { A } ph  /\ 
A. x  e.  { B } ph )  <->  ( ps  /\ 
ch ) ) )
104, 9syl5bb 257 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    u. cun 3474   {csn 4027   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-sbc 3332  df-un 3481  df-sn 4028  df-pr 4030
This theorem is referenced by:  raltpg  4078  ralpr  4080  iinxprg  4403  disjprg  4443  f12dfv  6167  f13dfv  6168  suppr  7929  injresinjlem  11893  wwlktovf1  12858  gcdcllem2  14009  joinval2lem  15495  meetval2lem  15509  iccntr  21089  limcun  22062  cusgra2v  24166  cusgra3v  24168  spthispth  24279  usgrcyclnl2  24345  4cycl4v4e  24370  4cycl4dv4e  24372  frgra3v  24706  3vfriswmgra  24709  sumpr  27459  prsiga  27799  usgra2pthlem1  31848  usgra2pth  31849
  Copyright terms: Public domain W3C validator