Theorem sbco3v 2020

Description: Version of sbco3 2025 with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbco3v	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Distinct variable group:   x, y

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

Proof of Theorem sbco3v

Step	Hyp	Ref	Expression
1		nfs1v 1990	. . . 4 |- F/ x [ y / x ] ph
2	1	nfri 1565	. . 3 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	2	sbco2vh 1996	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
4		sbco 2019	. . 3 |- ( [ x / y ] [ y / x ] ph <-> [ x / y ] ph )
5	4	sbbii 1811	. 2 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
6	3, 5	bitr3i 186	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-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:  sbcomv  2022