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Theorem raltpg 4083
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 4081 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
4 raltpg.3 . . . . 5  |-  ( x  =  C  ->  ( ph 
<->  th ) )
54ralsng 4067 . . . 4  |-  ( C  e.  X  ->  ( A. x  e.  { C } ph  <->  th ) )
63, 5bi2anan9 873 . . 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 1191 . 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 4037 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
98raleqi 3058 . . 3  |-  ( A. x  e.  { A ,  B ,  C } ph 
<-> 
A. x  e.  ( { A ,  B }  u.  { C } ) ph )
10 ralunb 3681 . . 3  |-  ( A. x  e.  ( { A ,  B }  u.  { C } )
ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
119, 10bitri 249 . 2  |-  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
12 df-3an 975 . 2  |-  ( ( ps  /\  ch  /\  th )  <->  ( ( ps 
/\  ch )  /\  th ) )
137, 11, 123bitr4g 288 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A. x  e. 
{ A ,  B ,  C } ph  <->  ( ps  /\ 
ch  /\  th )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807    u. cun 3469   {csn 4032   {cpr 4034   {ctp 4036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-sbc 3328  df-un 3476  df-sn 4033  df-pr 4035  df-tp 4037
This theorem is referenced by:  raltp  4087  raltpd  4155  f13dfv  6181  nb3grapr  24580  cusgra3v  24591  3v3e3cycl1  24771  constr3trllem2  24778  constr3trllem5  24781  frgra3v  25129
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