Theorem spcv 3299

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

Step	Hyp	Ref	Expression
1		spcv.1	. 2 |- A e. _V
2		spcv.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	spcgv 3293	. 2 |- ( A e. _V -> ( A. x ph -> ps ) )
4	1, 3	ax-mp 5	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  morex  3390  rext  4916  relop  5272  frxp  7287  pssnn  8178  findcard  8199  fiint  8237  marypha1lem  8339  dfom3  8544  elom3  8545  aceq3lem  8943  dfac3  8944  dfac5lem4  8949  dfac8  8957  dfac9  8958  dfacacn  8963  dfac13  8964  kmlem1  8972  kmlem10  8981  fin23lem34  9168  fin23lem35  9169  zorn2lem7  9324  zornn0g  9327  axgroth6  9650  nnunb  11288  symggen  17890  gsumval3lem2  18307  gsumzaddlem  18321  dfac14  21421  i1fd  23448  chlimi  28091  ddemeas  30299  dfpo2  31645  dfon2lem4  31691  dfon2lem5  31692  dfon2lem7  31694  ttac  37603  dfac11  37632  dfac21  37636  setrec2fun  42439