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Theorem rspc2ev 2856
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 2841 . . . 4  |-  ( ( B  e.  D  /\  ps )  ->  E. y  e.  D  ch )
32anim2i 342 . . 3  |-  ( ( A  e.  C  /\  ( B  e.  D  /\  ps ) )  -> 
( A  e.  C  /\  E. y  e.  D  ch ) )
433impb 1199 . 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 2478 . . 3  |-  ( x  =  A  ->  ( E. y  e.  D  ph  <->  E. y  e.  D  ch ) )
76rspcev 2841 . 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739
This theorem is referenced by:  rspc3ev  2858  opelxp  4656  rspceov  5916  2dom  6804  apreim  8559  addcn2  11317  mulcn2  11319  divalglemnn  11922  bezoutlema  11999  bezoutlemb  12000  pythagtriplem18  12280  pczpre  12296  pcdiv  12301  4sqlem3  12387  4sqlem4  12389  txuni2  13726  txopn  13735  txdis  13747  txdis1cn  13748  xmettxlem  13979  2irrexpq  14364  2irrexpqap  14366  2sqlem2  14432  2sqlem8  14440
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