Theorem sbnv 1876

Description: Version of sbn 1940 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 1874	. . 3 |- ( [ y / x ] -. ph <-> A. x ( x = y -> -. ph ) )
2		alinexa 1591	. . 3 |- ( A. x ( x = y -> -. ph ) <-> -. E. x ( x = y /\ ph ) )
3	1, 2	bitri 183	. 2 |- ( [ y / x ] -. ph <-> -. E. x ( x = y /\ ph ) )
4		sb5 1875	. 2 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
5	3, 4	xchbinxr 673	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   E.wex 1480   [wsb 1750

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-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-sb 1751

This theorem is referenced by:  sbn  1940