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Theorem rspcedv 2677
Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcdv.1  |-  ( ph  ->  A  e.  B )
rspcdv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rspcedv  |-  ( 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 rspcedv
StepHypRef Expression
1 rspcdv.1 . 2  |-  ( ph  ->  A  e.  B )
2 rspcdv.2 . . 3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
32biimprd 151 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
41, 3rspcimedv 2675 1  |-  ( ph  ->  ( ch  ->  E. x  e.  B  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   E.wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576
This theorem is referenced by:  rspcedvd  2680  rexxfrd  4223  ltexnqq  6564  halfnqq  6566  ltbtwnnqq  6571  genpml  6673  genpmu  6674  genprndl  6677  genprndu  6678  axarch  7023  apreap  7652
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