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

Theorem rspcdv 3217
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 207 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps  ->  ch ) )
41, 3rspcimdv 3215 1  |-  ( ph  ->  ( A. x  e.  B  ps  ->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115
This theorem is referenced by:  ralxfrd  4661  suppofss1d  6937  suppofss2d  6938  zindd  10962  wrd2ind  12666  ismri2dad  14892  mreexd  14897  mreexexlemd  14899  catcocl  14940  catass  14941  moni  14992  subccocl  15072  funcco  15098  fullfo  15139  fthf1  15144  nati  15182  mrcmndind  15816  idsrngd  17311  sizeusglecusglem1  24188  fargshiftfva  24343  wlkiswwlk2lem5  24399  usg2wlkeq  24412  clwlkisclwwlklem2a  24489  clwlkisclwwlklem1  24491  clwwisshclww  24511  usg2cwwk2dif  24524  eupatrl  24672  rngurd  27469  esumcvg  27760  orvcelel  28076  signsply0  28176  onint1  29519  ralbinrald  31699  ralxfrd2  31798
  Copyright terms: Public domain W3C validator