Theorem spcgv 2904

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 2384	. 2 |- F/_ x A
2		nfv 1577	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2899	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2203

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815

This theorem is referenced by:  spcv  2911  mob2  2997  intss1  3964  dfiin2g  4024  exmidsssnc  4316  exmid1stab  4321  frirrg  4471  frind  4473  alxfr  4582  elirr  4663  en2lp  4676  tfisi  4709  mptfvex  5763  tfrcl  6595  rdgisucinc  6616  frecabex  6629  fisseneq  7195  mkvprop  7449  exmidfodomrlemr  7505  exmidfodomrlemrALT  7506  acfun  7514  exmidmotap  7575  ccfunen  7578  zfz1isolem1  11212  zfz1iso  11213  uniopn  14866  pw1nct  16777  sbthom  16806