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

Theorem rextpg 3926
Description: Convert a quantification over a triple to a disjunction. (Contributed 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
rextpg  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. 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 rextpg
StepHypRef Expression
1 ralprg.1 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 ralprg.2 . . . . . 6  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
31, 2rexprg 3924 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
43orbi1d 702 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( E. x  e.  { A ,  B } ph  \/  E. x  e.  { C } ph ) 
<->  ( ( ps  \/  ch )  \/  E. x  e.  { C } ph ) ) )
5 raltpg.3 . . . . . 6  |-  ( x  =  C  ->  ( ph 
<->  th ) )
65rexsng 3911 . . . . 5  |-  ( C  e.  X  ->  ( E. x  e.  { C } ph  <->  th ) )
76orbi2d 701 . . . 4  |-  ( C  e.  X  ->  (
( ( ps  \/  ch )  \/  E. x  e.  { C } ph ) 
<->  ( ( ps  \/  ch )  \/  th )
) )
84, 7sylan9bb 699 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  (
( E. x  e. 
{ A ,  B } ph  \/  E. x  e.  { C } ph ) 
<->  ( ( ps  \/  ch )  \/  th )
) )
983impa 1182 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( E. x  e.  { A ,  B } ph  \/  E. x  e.  { C } ph ) 
<->  ( ( ps  \/  ch )  \/  th )
) )
10 df-tp 3880 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
1110rexeqi 2920 . . 3  |-  ( E. x  e.  { A ,  B ,  C } ph 
<->  E. x  e.  ( { A ,  B }  u.  { C } ) ph )
12 rexun 3534 . . 3  |-  ( E. x  e.  ( { A ,  B }  u.  { C } )
ph 
<->  ( E. x  e. 
{ A ,  B } ph  \/  E. x  e.  { C } ph ) )
1311, 12bitri 249 . 2  |-  ( E. x  e.  { A ,  B ,  C } ph 
<->  ( E. x  e. 
{ A ,  B } ph  \/  E. x  e.  { C } ph ) )
14 df-3or 966 . 2  |-  ( ( ps  \/  ch  \/  th )  <->  ( ( ps  \/  ch )  \/ 
th ) )
159, 13, 143bitr4g 288 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. x  e. 
{ A ,  B ,  C } ph  <->  ( ps  \/  ch  \/  th )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2714    u. cun 3324   {csn 3875   {cpr 3877   {ctp 3879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-v 2972  df-sbc 3185  df-un 3331  df-sn 3876  df-pr 3878  df-tp 3880
This theorem is referenced by:  rextp  3930  fr3nr  6389  nb3graprlem2  23358  frgra3vlem2  30590  3vfriswmgra  30594
  Copyright terms: Public domain W3C validator