ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralpr Unicode version

Theorem ralpr 3546
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 3542 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
61, 2, 5mp2an 420 1  |-  ( A. x  e.  { A ,  B } ph  <->  ( ps  /\ 
ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2391   _Vcvv 2658   {cpr 3496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-sbc 2881  df-un 3043  df-sn 3501  df-pr 3502
This theorem is referenced by:  fzprval  9802
  Copyright terms: Public domain W3C validator