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

Theorem raltpg 3861
Description: Convert a quantification over a triple 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 ) )
raltpg.3  |-  ( x  =  C  ->  ( ph 
<->  th ) )
Assertion
Ref Expression
raltpg  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x  e. 
{ A ,  B ,  C } ph  <->  ( ps  /\ 
ch  /\  th )
) )
Distinct variable groups:    x, A    x, B    x, C    ps, x    ch, x    th, x
Allowed substitution hints:    ph( x)    V( x)    W( x)    X( x)

Proof of Theorem raltpg
StepHypRef Expression
1 ralprg.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 ralprg.2 . . . . 5  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
31, 2ralprg 3859 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
4 raltpg.3 . . . . 5  |-  ( x  =  C  ->  ( ph 
<->  th ) )
54ralsng 3848 . . . 4  |-  ( C  e.  X  ->  ( A. x  e.  { C } ph  <->  th ) )
63, 5bi2anan9 845 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  (
( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) 
<->  ( ( ps  /\  ch )  /\  th )
) )
763impa 1149 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A. x  e.  { A ,  B } ph  /\  A. x  e.  { C } ph ) 
<->  ( ( ps  /\  ch )  /\  th )
) )
8 df-tp 3824 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
98raleqi 2910 . . 3  |-  ( A. x  e.  { A ,  B ,  C } ph 
<-> 
A. x  e.  ( { A ,  B }  u.  { C } ) ph )
10 ralunb 3530 . . 3  |-  ( A. x  e.  ( { A ,  B }  u.  { C } )
ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
119, 10bitri 242 . 2  |-  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
12 df-3an 939 . 2  |-  ( ( ps  /\  ch  /\  th )  <->  ( ( ps 
/\  ch )  /\  th ) )
137, 11, 123bitr4g 281 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x  e. 
{ A ,  B ,  C } ph  <->  ( ps  /\ 
ch  /\  th )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    u. cun 3320   {csn 3816   {cpr 3817   {ctp 3818
This theorem is referenced by:  raltp  3865  nb3grapr  21467  cusgra3v  21478  3v3e3cycl1  21636  constr3trllem2  21643  constr3trllem5  21646  f13dfv  28099  frgra3v  28466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-sbc 3164  df-un 3327  df-sn 3822  df-pr 3823  df-tp 3824
  Copyright terms: Public domain W3C validator