Theorem spcgv 2707

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

Hypothesis

Ref	Expression
spcgv.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
spcgv	|- ( A e. V -> ( A. x ph -> ps ) )

Distinct variable groups:   ps, x   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem spcgv

Step	Hyp	Ref	Expression
1		nfcv 2229	. 2 |- F/_ x A
2		nfv 1467	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2702	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1288   = wceq 1290   e. wcel 1439

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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622

This theorem is referenced by:  spcv  2713  mob2  2796  intss1  3709  dfiin2g  3769  frirrg  4186  frind  4188  alxfr  4296  elirr  4370  en2lp  4383  tfisi  4415  mptfvex  5401  tfrcl  6143  rdgisucinc  6164  frecabex  6177  fisseneq  6696  exmidfodomrlemr  6889  exmidfodomrlemrALT  6890  zfz1isolem1  10306  zfz1iso  10307  uniopn  11761