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Theorem rexprg 3683
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 3647 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
21rexeqi 2741 . . 3  |-  ( E. x  e.  { A ,  B } ph  <->  E. x  e.  ( { A }  u.  { B } )
ph )
3 rexun 3355 . . 3  |-  ( E. x  e.  ( { A }  u.  { B } ) ph  <->  ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph ) )
42, 3bitri 240 . 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 3673 . . . 4  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
76orbi1d 683 . . 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 3673 . . . 4  |-  ( B  e.  W  ->  ( E. x  e.  { B } ph  <->  ch ) )
109orbi2d 682 . . 3  |-  ( B  e.  W  ->  (
( ps  \/  E. x  e.  { B } ph )  <->  ( ps  \/  ch ) ) )
117, 10sylan9bb 680 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( E. x  e.  { A } ph  \/  E. x  e.  { B } ph )  <->  ( ps  \/  ch ) ) )
124, 11syl5bb 248 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    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    u. cun 3150   {csn 3640   {cpr 3641
This theorem is referenced by:  rextpg  3685  rexpr  3687  fr2nr  4371  frgra2v  27539  3vfriswmgralem  27544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647
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