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Theorem spcv 2646
Description: Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)
Hypotheses
Ref Expression
spcv.1  |-  A  e. 
_V
spcv.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spcv  |-  ( A. x ph  ->  ps )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem spcv
StepHypRef Expression
1 spcv.1 . 2  |-  A  e. 
_V
2 spcv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32spcgv 2640 . 2  |-  ( A  e.  _V  ->  ( A. x ph  ->  ps ) )
41, 3ax-mp 7 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   _Vcvv 2557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by:  morex  2725  rext  3951  ontr2exmid  4250  regexmidlem1  4258  reg2exmid  4261  relop  4486  rdgtfr  5961  ssfiexmid  6336  diffitest  6344  findcard  6345  bj-d0clsepcl  10049  bj-inf2vnlem1  10095
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