Theorem sbcie 2848

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)

Hypotheses

Ref	Expression
sbcie.1	|- A e. _V
sbcie.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbcie	|- ( [. A / x ]. ph <-> ps )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem sbcie

Step	Hyp	Ref	Expression
1		sbcie.1	. 2 |- A e. _V
2		sbcie.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 2846	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	ax-mp 7	1 |- ( [. A / x ]. ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   [.wsbc 2815

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-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  findcard2  6373  findcard2s  6374  ac6sfi  6379  nn1suc  8058  indstr  8681  bezoutlemmain  10387  prmind2  10502