Theorem spcv 2692

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 2686	. 2 |- ( A e. _V -> ( A. x ph -> ps ) )
4	1, 3	ax-mp 7	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283   = wceq 1285   e. wcel 1434   _Vcvv 2602

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604

This theorem is referenced by:  morex  2777  rext  3978  ontr2exmid  4276  regexmidlem1  4284  reg2exmid  4287  relop  4514  rdgtfr  6023  ssfiexmid  6411  domfiexmid  6413  diffitest  6421  findcard  6422  bj-d0clsepcl  10878  bj-inf2vnlem1  10923