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Theorem raltpg 3774
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 3772 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
4 raltpg.3 . . . . 5  |-  ( x  =  C  ->  ( ph 
<->  th ) )
54ralsng 3761 . . . 4  |-  ( C  e.  X  ->  ( A. x  e.  { C } ph  <->  th ) )
63, 5bi2anan9 843 . . 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 1147 . 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 3737 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
98raleqi 2825 . . 3  |-  ( A. x  e.  { A ,  B ,  C } ph 
<-> 
A. x  e.  ( { A ,  B }  u.  { C } ) ph )
10 ralunb 3444 . . 3  |-  ( A. x  e.  ( { A ,  B }  u.  { C } )
ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
119, 10bitri 240 . 2  |-  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
12 df-3an 937 . 2  |-  ( ( ps  /\  ch  /\  th )  <->  ( ( ps 
/\  ch )  /\  th ) )
137, 11, 123bitr4g 279 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    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628    u. cun 3236   {csn 3729   {cpr 3730   {ctp 3731
This theorem is referenced by:  raltp  3778  nb3grapr  27618  cusgra3v  27629  3v3e3cycl1  27770  constr3trllem2  27777  constr3trllem5  27780  frgra3v  27837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-v 2875  df-sbc 3078  df-un 3243  df-sn 3735  df-pr 3736  df-tp 3737
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