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Theorem rexprg 4077
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 4030 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
21rexeqi 3063 . . 3  |-  ( E. x  e.  { A ,  B } ph  <->  E. x  e.  ( { A }  u.  { B } )
ph )
3 rexun 3684 . . 3  |-  ( E. x  e.  ( { A }  u.  { B } ) ph  <->  ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph ) )
42, 3bitri 249 . 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 4063 . . . 4  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
76orbi1d 702 . . 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 4063 . . . 4  |-  ( B  e.  W  ->  ( E. x  e.  { B } ph  <->  ch ) )
109orbi2d 701 . . 3  |-  ( B  e.  W  ->  (
( ps  \/  E. x  e.  { B } ph )  <->  ( ps  \/  ch ) ) )
117, 10sylan9bb 699 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph )  <->  ( ps  \/  ch ) ) )
124, 11syl5bb 257 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x  e. 
{ A ,  B } ph  <->  ( ps  \/  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    u. cun 3474   {csn 4027   {cpr 4029
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-rex 2820  df-v 3115  df-sbc 3332  df-un 3481  df-sn 4028  df-pr 4030
This theorem is referenced by:  rextpg  4079  rexpr  4081  fr2nr  4857  nb3graprlem2  24225  frgra2v  24772  3vfriswmgralem  24777  ldepspr  32372  zlmodzxzldeplem4  32402
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