Theorem sbcomv 1894

Description: Version of sbcom 1898 with a distinct variable constraint between x and z. (Contributed by Jim Kingdon, 28-Feb-2018.)

Assertion

Ref	Expression
sbcomv	|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )

Distinct variable group:   x, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem sbcomv

Step	Hyp	Ref	Expression
1		sbco3v 1892	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] [ x / z ] ph )
2		sbcocom 1893	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] [ y / x ] ph )
3		sbcocom 1893	. 2 |- ( [ y / x ] [ x / z ] ph <-> [ y / x ] [ y / z ] ph )
4	1, 2, 3	3bitr3i 209	1 |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )

Syntax hints:   <-> wb 104   [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694

This theorem is referenced by: (None)