Theorem spcgv 2773

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 2281	. 2 |- F/_ x A
2		nfv 1508	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2768	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  spcv  2779  mob2  2864  intss1  3786  dfiin2g  3846  exmidsssnc  4126  frirrg  4272  frind  4274  alxfr  4382  elirr  4456  en2lp  4469  tfisi  4501  mptfvex  5506  tfrcl  6261  rdgisucinc  6282  frecabex  6295  fisseneq  6820  mkvprop  7032  exmidfodomrlemr  7058  exmidfodomrlemrALT  7059  acfun  7063  ccfunen  7079  zfz1isolem1  10583  zfz1iso  10584  uniopn  12168  exmid1stab  13195  sbthom  13221