Theorem sbcomv 1845

Description: Version of sbcom 1849 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 1843	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] [ x / z ] ph )
2		sbcocom 1844	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] [ y / x ] ph )
3		sbcocom 1844	. 2 |- ( [ y / x ] [ x / z ] ph <-> [ y / x ] [ y / z ] ph )
4	1, 2, 3	3bitr3i 199	1 |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )

Syntax hints:   <-> wb 98   [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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)