MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspcdv Unicode version

Theorem rspcdv 2887
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-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
rspcdv  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Distinct variable groups:    x, A    x, B    ph, x    ch, x
Allowed substitution hint:    ps( x)

Proof of Theorem rspcdv
StepHypRef Expression
1 rspcdv.1 . 2  |-  ( ph  ->  A  e.  B )
2 rspcdv.2 . . 3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
32biimpd 198 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
41, 3rspcimdv 2885 1  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543
This theorem is referenced by:  ralxfrd  4548  zindd  10113  ismri2dad  13539  mreexd  13544  mreexexlemd  13546  catcocl  13587  catass  13588  moni  13639  subccocl  13719  funcco  13745  fullfo  13786  fthf1  13791  nati  13829  esumcvg  23454  orvcelel  23670  onint1  24888  imonclem  25813  iepiclem  25823  ralbinrald  27977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790
  Copyright terms: Public domain W3C validator