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Theorem raltpg 3490
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 3488 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
4 raltpg.3 . . . . 5  |-  ( x  =  C  ->  ( ph 
<->  th ) )
54ralsng 3478 . . . 4  |-  ( C  e.  X  ->  ( A. x  e.  { C } ph  <->  th ) )
63, 5bi2anan9 573 . . 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 1138 . 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 3449 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
98raleqi 2566 . . 3  |-  ( A. x  e.  { A ,  B ,  C } ph 
<-> 
A. x  e.  ( { A ,  B }  u.  { C } ) ph )
10 ralunb 3179 . . 3  |-  ( A. x  e.  ( { A ,  B }  u.  { C } )
ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
119, 10bitri 182 . 2  |-  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
12 df-3an 926 . 2  |-  ( ( ps  /\  ch  /\  th )  <->  ( ( ps 
/\  ch )  /\  th ) )
137, 11, 123bitr4g 221 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    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359    u. cun 2995   {csn 3441   {cpr 3442   {ctp 3443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448  df-tp 3449
This theorem is referenced by:  raltp  3494  sumtp  10771
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