Theorem sbcie 3079

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 3077	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	ax-mp 5	1 |- ( [. A / x ]. ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   [.wsbc 3044

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-sbc 3045

This theorem is referenced by:  findcard2  7148  findcard2s  7149  ac6sfi  7157  nn1suc  9258  indstr  9928  wrdind  11418  bezoutlemmain  12698  prmind2  12821  isghm  13977