Theorem sbcie 3024

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 3022	. 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 1364   e. wcel 2167   _Vcvv 2763   [.wsbc 2989

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  findcard2  6950  findcard2s  6951  ac6sfi  6959  nn1suc  9009  indstr  9667  bezoutlemmain  12165  prmind2  12288  isghm  13373