Theorem sbnv 1809

Description: Version of sbn 1867 where x and y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)

Assertion

Ref	Expression
sbnv	|- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem sbnv

Step	Hyp	Ref	Expression
1		sb6 1807	. . 3 |- ( [ y / x ] -. ph <-> A. x ( x = y -> -. ph ) )
2		alinexa 1534	. . 3 |- ( A. x ( x = y -> -. ph ) <-> -. E. x ( x = y /\ ph ) )
3	1, 2	bitri 182	. 2 |- ( [ y / x ] -. ph <-> -. E. x ( x = y /\ ph ) )
4		sb5 1808	. 2 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
5	3, 4	xchbinxr 640	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   [wsb 1685

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-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-sb 1686

This theorem is referenced by:  sbn  1867