Theorem sbcie 2943

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 2941	. 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 1331   e. wcel 1480   _Vcvv 2686   [.wsbc 2909

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910

This theorem is referenced by:  findcard2  6783  findcard2s  6784  ac6sfi  6792  nn1suc  8739  indstr  9388  bezoutlemmain  11686  prmind2  11801