Theorem spcv 2779

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 2773	. 2 |- ( A e. _V -> ( A. x ph -> ps ) )
4	1, 3	ax-mp 5	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480   _Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  morex  2868  exmidexmid  4120  exmidsssn  4125  exmidel  4128  rext  4137  ontr2exmid  4440  regexmidlem1  4448  reg2exmid  4451  relop  4689  disjxp1  6133  rdgtfr  6271  ssfiexmid  6770  domfiexmid  6772  diffitest  6781  findcard  6782  fiintim  6817  fisseneq  6820  finomni  7012  exmidomni  7014  exmidlpo  7015  exmidunben  11939  bj-d0clsepcl  13123  bj-inf2vnlem1  13168  subctctexmid  13196