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Theorem rextpg 3572
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 3570 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
43orbi1d 780 . . . 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 3560 . . . . 5  |-  ( C  e.  X  ->  ( E. x  e.  { C } ph  <->  th ) )
76orbi2d 779 . . . 4  |-  ( C  e.  X  ->  (
( ( ps  \/  ch )  \/  E. x  e.  { C } ph ) 
<->  ( ( ps  \/  ch )  \/  th )
) )
84, 7sylan9bb 457 . . 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 1176 . 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 3530 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
1110rexeqi 2629 . . 3  |-  ( E. x  e.  { A ,  B ,  C } ph 
<->  E. x  e.  ( { A ,  B }  u.  { C } ) ph )
12 rexun 3251 . . 3  |-  ( E. x  e.  ( { A ,  B }  u.  { C } )
ph 
<->  ( E. x  e. 
{ A ,  B } ph  \/  E. x  e.  { C } ph ) )
1311, 12bitri 183 . 2  |-  ( E. x  e.  { A ,  B ,  C } ph 
<->  ( E. x  e. 
{ A ,  B } ph  \/  E. x  e.  { C } ph ) )
14 df-3or 963 . 2  |-  ( ( ps  \/  ch  \/  th )  <->  ( ( ps  \/  ch )  \/ 
th ) )
159, 13, 143bitr4g 222 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    /\ wa 103    <-> wb 104    \/ wo 697    \/ w3o 961    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2415    u. cun 3064   {csn 3522   {cpr 3523   {ctp 3524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-sn 3528  df-pr 3529  df-tp 3530
This theorem is referenced by:  rextp  3576
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