Theorem spcgv 2685

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 2219	. 2 |- F/_ x A
2		nfv 1461	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcgf 2680	1 |- ( A e. V -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  spcv  2691  mob2  2772  intss1  3651  dfiin2g  3711  frirrg  4105  frind  4107  alxfr  4211  elirr  4284  en2lp  4297  tfisi  4328  mptfvex  5277  rdgisucinc  5995  frecabex  6007