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Theorem raltpg 4025
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 4023 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
4 raltpg.3 . . . . 5  |-  ( x  =  C  ->  ( ph 
<->  th ) )
54ralsng 4008 . . . 4  |-  ( C  e.  X  ->  ( A. x  e.  { C } ph  <->  th ) )
63, 5bi2anan9 885 . . 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 1204 . 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 3975 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
98raleqi 2993 . . 3  |-  ( A. x  e.  { A ,  B ,  C } ph 
<-> 
A. x  e.  ( { A ,  B }  u.  { C } ) ph )
10 ralunb 3617 . . 3  |-  ( A. x  e.  ( { A ,  B }  u.  { C } )
ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
119, 10bitri 253 . 2  |-  ( A. x  e.  { A ,  B ,  C } ph 
<->  ( A. x  e. 
{ A ,  B } ph  /\  A. x  e.  { C } ph ) )
12 df-3an 988 . 2  |-  ( ( ps  /\  ch  /\  th )  <->  ( ( ps 
/\  ch )  /\  th ) )
137, 11, 123bitr4g 292 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739    u. cun 3404   {csn 3970   {cpr 3972   {ctp 3974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-v 3049  df-sbc 3270  df-un 3411  df-sn 3971  df-pr 3973  df-tp 3975
This theorem is referenced by:  raltp  4029  raltpd  4098  f13dfv  6178  sumtp  13822  lcmftp  14621  nb3grapr  25193  cusgra3v  25204  3v3e3cycl1  25384  constr3trllem2  25391  constr3trllem5  25394  frgra3v  25742  nb3grpr  39466
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