Theorem sbnv 1768

Description: Version of sbn 1826 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 1766	. . 3 |- ( [ y / x ] -. ph <-> A. x ( x = y -> -. ph ) )
2		alinexa 1494	. . 3 |- ( A. x ( x = y -> -. ph ) <-> -. E. x ( x = y /\ ph ) )
3	1, 2	bitri 173	. 2 |- ( [ y / x ] -. ph <-> -. E. x ( x = y /\ ph ) )
4		sb5 1767	. 2 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
5	3, 4	xchbinxr 608	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-sb 1646

This theorem is referenced by:  sbn  1826