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Theorem rspcimedv 2763
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 109 . . . . . . 7  |-  ( (
ph  /\  x  =  A )  ->  x  =  A )
32eleq1d 2184 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  (
x  e.  B  <->  A  e.  B ) )
43biimprd 157 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( A  e.  B  ->  x  e.  B ) )
5 rspcimedv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
64, 5anim12d 331 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  (
( A  e.  B  /\  ch )  ->  (
x  e.  B  /\  ps ) ) )
71, 6spcimedv 2744 . . 3  |-  ( ph  ->  ( ( A  e.  B  /\  ch )  ->  E. x ( x  e.  B  /\  ps ) ) )
81, 7mpand 423 . 2  |-  ( ph  ->  ( ch  ->  E. x
( x  e.  B  /\  ps ) ) )
9 df-rex 2397 . 2  |-  ( E. x  e.  B  ps  <->  E. x ( x  e.  B  /\  ps )
)
108, 9syl6ibr 161 1  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   E.wex 1451    e. wcel 1463   E.wrex 2392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 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-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660
This theorem is referenced by:  rspcedv  2765
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