MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cla4gv Unicode version

Theorem cla4gv 2843
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)
Hypothesis
Ref Expression
cla4gv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cla4gv  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem cla4gv
StepHypRef Expression
1 nfcv 2394 . 2  |-  F/_ x A
2 nfv 1629 . 2  |-  F/ x ps
3 cla4gv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3cla4gf 2838 1  |-  ( A  e.  V  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532    = wceq 1619    e. wcel 1621
This theorem is referenced by:  cla4v  2849  mob2  2920  intss1  3851  dfiin2g  3910  fri  4327  alxfr  4519  tfisi  4621  limomss  4633  nnlim  4641  isofrlem  5771  f1oweALT  5785  pssnn  7049  findcard3  7068  ttukeylem1  8104  rami  13025  ramcl  13039  clatlem  14179  islbs3  15871  mplsubglem  16142  mpllsslem  16143  uniopn  16606  chlimi  21775  relexpind  23410  dfon2lem3  23511  dfon2lem8  23516  sgplpte21b  25502  neificl  25835  ismrcd1  26141
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-v 2765
  Copyright terms: Public domain W3C validator