Theorem sbco3xzyz 1847

Description: Version of sbco3 1848 with distinct variable constraints between x and z, and y and z. Lemma for proving sbco3 1848. (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 1846	. 2 |- ( [ z / y ] [ z / x ] ph <-> [ z / x ] [ z / y ] ph )
2		sbcocom 1844	. 2 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )
3		sbcocom 1844	. 2 |- ( [ z / x ] [ x / y ] ph <-> [ z / x ] [ z / y ] ph )
4	1, 2, 3	3bitr4i 201	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] 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-bndl 1399  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:  sbco3  1848