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Theorem rexprg 3489
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 3448 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
21rexeqi 2567 . . 3  |-  ( E. x  e.  { A ,  B } ph  <->  E. x  e.  ( { A }  u.  { B } )
ph )
3 rexun 3178 . . 3  |-  ( E. x  e.  ( { A }  u.  { B } ) ph  <->  ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph ) )
42, 3bitri 182 . 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 3479 . . . 4  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
76orbi1d 740 . . 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 3479 . . . 4  |-  ( B  e.  W  ->  ( E. x  e.  { B } ph  <->  ch ) )
109orbi2d 739 . . 3  |-  ( B  e.  W  ->  (
( ps  \/  E. x  e.  { B } ph )  <->  ( ps  \/  ch ) ) )
117, 10sylan9bb 450 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph )  <->  ( ps  \/  ch ) ) )
124, 11syl5bb 190 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 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   E.wrex 2360    u. cun 2995   {csn 3441   {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by:  rextpg  3491  rexpr  3493  minmax  10624
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