Theorem sbco3xzyz 2024

Description: Version of sbco3 2025 with distinct variable constraints between x and z, and y and z. Lemma for proving sbco3 2025. (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 2023	. 2 |- ( [ z / y ] [ z / x ] ph <-> [ z / x ] [ z / y ] ph )
2		sbcocom 2021	. 2 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )
3		sbcocom 2021	. 2 |- ( [ z / x ] [ x / y ] ph <-> [ z / x ] [ z / y ] ph )
4	1, 2, 3	3bitr4i 212	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 105   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sbco3  2025