Theorem sbnv 1903

Description: Version of sbn 1971 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 1901	. . 3 |- ( [ y / x ] -. ph <-> A. x ( x = y -> -. ph ) )
2		alinexa 1617	. . 3 |- ( A. x ( x = y -> -. ph ) <-> -. E. x ( x = y /\ ph ) )
3	1, 2	bitri 184	. 2 |- ( [ y / x ] -. ph <-> -. E. x ( x = y /\ ph ) )
4		sb5 1902	. 2 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
5	3, 4	xchbinxr 684	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1506   [wsb 1776

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-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-sb 1777

This theorem is referenced by:  sbn  1971