Theorem sbco3xzyz 1961

Description: Version of sbco3 1962 with distinct variable constraints between x and z, and y and z. Lemma for proving sbco3 1962. (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 1960	. 2 |- ( [ z / y ] [ z / x ] ph <-> [ z / x ] [ z / y ] ph )
2		sbcocom 1958	. 2 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )
3		sbcocom 1958	. 2 |- ( [ z / x ] [ x / y ] ph <-> [ z / x ] [ z / y ] ph )
4	1, 2, 3	3bitr4i 211	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 104   [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sbco3  1962