Theorem spcv 2748

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 2742	. 2 |- ( A e. _V -> ( A. x ph -> ps ) )
4	1, 3	ax-mp 7	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1310   = wceq 1312   e. wcel 1461   _Vcvv 2655

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657

This theorem is referenced by:  morex  2835  exmidexmid  4078  exmidsssn  4083  exmidel  4086  rext  4095  ontr2exmid  4398  regexmidlem1  4406  reg2exmid  4409  relop  4647  disjxp1  6085  rdgtfr  6223  ssfiexmid  6721  domfiexmid  6723  diffitest  6732  findcard  6733  fiintim  6768  fisseneq  6771  finomni  6960  exmidomni  6962  exmidlpo  6963  exmidunben  11778  bj-d0clsepcl  12806  bj-inf2vnlem1  12851