Theorem sbnv 1888

Description: Version of sbn 1952 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 1886	. . 3 |- ( [ y / x ] -. ph <-> A. x ( x = y -> -. ph ) )
2		alinexa 1603	. . 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 1887	. 2 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
5	3, 4	xchbinxr 683	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   E.wex 1492   [wsb 1762

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-sb 1763

This theorem is referenced by:  sbn  1952