Theorem spcv 2897

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 2890	. 2 |- ( A e. _V -> ( A. x ph -> ps ) )
4	1, 3	ax-mp 5	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   _Vcvv 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  morex  2987  exmidexmid  4280  exmidsssn  4286  exmidel  4289  rext  4301  ontr2exmid  4617  regexmidlem1  4625  reg2exmid  4628  relop  4872  uchoice  6289  disjxp1  6388  rdgtfr  6526  ssfiexmid  7046  domfiexmid  7048  diffitest  7057  findcard  7058  exmidpw2en  7085  fiintim  7104  fisseneq  7107  finomni  7318  exmidomni  7320  exmidlpo  7321  exmidunben  13013  ivthreinc  15335  bj-d0clsepcl  16371  bj-inf2vnlem1  16416  subctctexmid  16453