Theorem spcgv 2894

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 2375	. 2 |- F/_ x A
2		nfv 1577	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2889	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   = wceq 1398   e. wcel 2202

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  spcv  2901  mob2  2987  intss1  3948  dfiin2g  4008  exmidsssnc  4299  exmid1stab  4304  frirrg  4453  frind  4455  alxfr  4564  elirr  4645  en2lp  4658  tfisi  4691  mptfvex  5741  tfrcl  6573  rdgisucinc  6594  frecabex  6607  fisseneq  7170  mkvprop  7417  exmidfodomrlemr  7473  exmidfodomrlemrALT  7474  acfun  7482  exmidmotap  7540  ccfunen  7543  zfz1isolem1  11167  zfz1iso  11168  uniopn  14812  pw1nct  16725  sbthom  16754