Theorem sbcie 2989

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 2987	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	ax-mp 5	1 |- ( [. A / x ]. ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   [.wsbc 2955

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956

This theorem is referenced by:  findcard2  6867  findcard2s  6868  ac6sfi  6876  nn1suc  8897  indstr  9552  bezoutlemmain  11953  prmind2  12074