Theorem spcgv 2890

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 2372	. 2 |- F/_ x A
2		nfv 1574	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2885	1 |- ( A e. V -> ( A. x ph -> ps ) )

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

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:  spcv  2897  mob2  2983  intss1  3938  dfiin2g  3998  exmidsssnc  4287  exmid1stab  4292  frirrg  4441  frind  4443  alxfr  4552  elirr  4633  en2lp  4646  tfisi  4679  mptfvex  5720  tfrcl  6510  rdgisucinc  6531  frecabex  6544  fisseneq  7096  mkvprop  7325  exmidfodomrlemr  7380  exmidfodomrlemrALT  7381  acfun  7389  exmidmotap  7447  ccfunen  7450  zfz1isolem1  11062  zfz1iso  11063  uniopn  14675  pw1nct  16369  sbthom  16394