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Theorem rextpg 3698
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 3696 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
43orbi1d 683 . . . 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 3686 . . . . 5  |-  ( C  e.  X  ->  ( E. x  e.  { C } ph  <->  th ) )
76orbi2d 682 . . . 4  |-  ( C  e.  X  ->  (
( ( ps  \/  ch )  \/  E. x  e.  { C } ph ) 
<->  ( ( ps  \/  ch )  \/  th )
) )
84, 7sylan9bb 680 . . 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 1146 . 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 3661 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
1110rexeqi 2754 . . 3  |-  ( E. x  e.  { A ,  B ,  C } ph 
<->  E. x  e.  ( { A ,  B }  u.  { C } ) ph )
12 rexun 3368 . . 3  |-  ( E. x  e.  ( { A ,  B }  u.  { C } )
ph 
<->  ( E. x  e. 
{ A ,  B } ph  \/  E. x  e.  { C } ph ) )
1311, 12bitri 240 . 2  |-  ( E. x  e.  { A ,  B ,  C } ph 
<->  ( E. x  e. 
{ A ,  B } ph  \/  E. x  e.  { C } ph ) )
14 df-3or 935 . 2  |-  ( ( ps  \/  ch  \/  th )  <->  ( ( ps  \/  ch )  \/ 
th ) )
159, 13, 143bitr4g 279 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( E. x  e. 
{ A ,  B ,  C } ph  <->  ( ps  \/  ch  \/  th )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557    u. cun 3163   {csn 3653   {cpr 3654   {ctp 3655
This theorem is referenced by:  rextp  3702  fr3nr  4587  nb3graprlem2  28288  frgra3vlem2  28425  3vfriswmgra  28429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-v 2803  df-sbc 3005  df-un 3170  df-sn 3659  df-pr 3660  df-tp 3661
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