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Theorem ralpr 4086
Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralpr.1  |-  A  e. 
_V
ralpr.2  |-  B  e. 
_V
ralpr.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ralpr.4  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
ralpr  |-  ( A. x  e.  { A ,  B } ph  <->  ( ps  /\ 
ch ) )
Distinct variable groups:    x, A    x, B    ps, x    ch, x
Allowed substitution hint:    ph( x)

Proof of Theorem ralpr
StepHypRef Expression
1 ralpr.1 . 2  |-  A  e. 
_V
2 ralpr.2 . 2  |-  B  e. 
_V
3 ralpr.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 ralpr.4 . . 3  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
53, 4ralprg 4082 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
61, 2, 5mp2an 672 1  |-  ( A. x  e.  { A ,  B } ph  <->  ( ps  /\ 
ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   {cpr 4035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-v 3120  df-sbc 3337  df-un 3486  df-sn 4034  df-pr 4036
This theorem is referenced by:  fzprval  11752  xpsfrnel  14835  xpsle  14853  isdrs2  15443  pmtrsn  16417  iblcnlem1  22062  wlkntrllem2  24385  wlkntrllem3  24386  2wlklem  24389  usgra2wlkspthlem1  24442  numclwwlkovf2ex  24910  subfacp1lem3  28451  fprb  29130  usgra2pthspth  32141  ldepsnlinc  32591
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