ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rspc2ev Unicode version

Theorem rspc2ev 2774
Description: 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
Hypotheses
Ref Expression
rspc2v.1  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
rspc2v.2  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
Assertion
Ref Expression
rspc2ev  |-  ( ( A  e.  C  /\  B  e.  D  /\  ps )  ->  E. x  e.  C  E. y  e.  D  ph )
Distinct variable groups:    x, y, A   
y, B    x, C    x, D, y    ch, x    ps, y
Allowed substitution hints:    ph( x, y)    ps( x)    ch( y)    B( x)    C( y)

Proof of Theorem rspc2ev
StepHypRef Expression
1 rspc2v.2 . . . . 5  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
21rspcev 2760 . . . 4  |-  ( ( B  e.  D  /\  ps )  ->  E. y  e.  D  ch )
32anim2i 337 . . 3  |-  ( ( A  e.  C  /\  ( B  e.  D  /\  ps ) )  -> 
( A  e.  C  /\  E. y  e.  D  ch ) )
433impb 1160 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  ps )  ->  ( A  e.  C  /\  E. y  e.  D  ch ) )
5 rspc2v.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
65rexbidv 2412 . . 3  |-  ( x  =  A  ->  ( E. y  e.  D  ph  <->  E. y  e.  D  ch ) )
76rspcev 2760 . 2  |-  ( ( A  e.  C  /\  E. y  e.  D  ch )  ->  E. x  e.  C  E. y  e.  D  ph )
84, 7syl 14 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  ps )  ->  E. x  e.  C  E. y  e.  D  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 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 2244  df-rex 2396  df-v 2659
This theorem is referenced by:  rspc3ev  2776  opelxp  4529  rspceov  5767  2dom  6653  apreim  8283  addcn2  10971  mulcn2  10973  divalglemnn  11463  bezoutlema  11533  bezoutlemb  11534  txuni2  12267  txopn  12276  txdis  12288  txdis1cn  12289  xmettxlem  12498
  Copyright terms: Public domain W3C validator