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

Theorem rexprg 3575
Description: Convert a quantification over a pair to a disjunction. (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 ) )
Assertion
Ref Expression
rexprg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
Distinct variable groups:    x, A    x, B    ps, x    ch, x
Allowed substitution hints:    ph( x)    V( x)    W( x)

Proof of Theorem rexprg
StepHypRef Expression
1 df-pr 3534 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
21rexeqi 2631 . . 3  |-  ( E. x  e.  { A ,  B } ph  <->  E. x  e.  ( { A }  u.  { B } )
ph )
3 rexun 3256 . . 3  |-  ( E. x  e.  ( { A }  u.  { B } ) ph  <->  ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph ) )
42, 3bitri 183 . 2  |-  ( E. x  e.  { A ,  B } ph  <->  ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph ) )
5 ralprg.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
65rexsng 3565 . . . 4  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
76orbi1d 780 . . 3  |-  ( A  e.  V  ->  (
( E. x  e. 
{ A } ph  \/  E. x  e.  { B } ph )  <->  ( ps  \/  E. x  e.  { B } ph ) ) )
8 ralprg.2 . . . . 5  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
98rexsng 3565 . . . 4  |-  ( B  e.  W  ->  ( E. x  e.  { B } ph  <->  ch ) )
109orbi2d 779 . . 3  |-  ( B  e.  W  ->  (
( ps  \/  E. x  e.  { B } ph )  <->  ( ps  \/  ch ) ) )
117, 10sylan9bb 457 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph )  <->  ( ps  \/  ch ) ) )
124, 11syl5bb 191 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   E.wrex 2417    u. cun 3069   {csn 3527   {cpr 3528
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534
This theorem is referenced by:  rextpg  3577  rexpr  3579  minmax  11008  xrminmax  11041
  Copyright terms: Public domain W3C validator