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Theorem rspcimedv 2724
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1  |-  ( ph  ->  A  e.  B )
rspcimedv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
Assertion
Ref Expression
rspcimedv  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Distinct variable groups:    x, A    x, B    ph, x    ch, x
Allowed substitution hint:    ps( x)

Proof of Theorem rspcimedv
StepHypRef Expression
1 rspcimdv.1 . . 3  |-  ( ph  ->  A  e.  B )
2 simpr 108 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
32eleq1d 2156 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
x  e.  B  <->  A  e.  B ) )
43biimprd 156 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( A  e.  B  ->  x  e.  B ) )
5 rspcimedv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
64, 5anim12d 328 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( A  e.  B  /\  ch )  ->  (
x  e.  B  /\  ps ) ) )
71, 6spcimedv 2705 . . 3  |-  ( ph  ->  ( ( A  e.  B  /\  ch )  ->  E. x ( x  e.  B  /\  ps ) ) )
81, 7mpand 420 . 2  |-  ( ph  ->  ( ch  ->  E. x
( x  e.  B  /\  ps ) ) )
9 df-rex 2365 . 2  |-  ( E. x  e.  B  ps  <->  E. x ( x  e.  B  /\  ps )
)
108, 9syl6ibr 160 1  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   E.wrex 2360
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-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
This theorem is referenced by:  rspcedv  2726
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