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

Theorem spcgv 2868
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
Hypothesis
Ref Expression
spcgv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spcgv  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem spcgv
StepHypRef Expression
1 nfcv 2419 . 2  |-  F/_ x A
2 nfv 1605 . 2  |-  F/ x ps
3 spcgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3spcgf 2863 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    = wceq 1623    e. wcel 1684
This theorem is referenced by:  spcv  2874  mob2  2945  intss1  3877  dfiin2g  3936  fri  4355  alxfr  4547  tfisi  4649  limomss  4661  nnlim  4669  isofrlem  5837  f1oweALT  5851  pssnn  7081  findcard3  7100  ttukeylem1  8136  rami  13062  ramcl  13076  clatlem  14216  islbs3  15908  mplsubglem  16179  mpllsslem  16180  uniopn  16643  chlimi  21814  relexpind  24037  dfon2lem3  24141  dfon2lem8  24146  sgplpte21b  26134  neificl  26467  ismrcd1  26773
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-v 2790
  Copyright terms: Public domain W3C validator