Theorem sbco3xzyz 1888

Description: Version of sbco3 1889 with distinct variable constraints between x and z, and y and z. Lemma for proving sbco3 1889. (Contributed by Jim Kingdon, 22-Mar-2018.)

Assertion

Ref	Expression
sbco3xzyz	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Distinct variable groups:   x, z   y, z

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

Proof of Theorem sbco3xzyz

Step	Hyp	Ref	Expression
1		sbcomxyyz 1887	. 2 |- ( [ z / y ] [ z / x ] ph <-> [ z / x ] [ z / y ] ph )
2		sbcocom 1885	. 2 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )
3		sbcocom 1885	. 2 |- ( [ z / x ] [ x / y ] ph <-> [ z / x ] [ z / y ] ph )
4	1, 2, 3	3bitr4i 210	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 103   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbco3  1889