Theorem sbco4 2466

Description: Two ways of exchanging two variables. Both sides of the biconditional exchange x and y, either via two temporary variables u and v, or a single temporary w. (Contributed by Jim Kingdon, 25-Sep-2018.)

Assertion

Ref	Expression
sbco4	|- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )

Distinct variable groups:   v, u, ph   x, u, v   y, u, v   ph, w   x, w   y, w

Allowed substitution hints:   ph( x, y)

Proof of Theorem sbco4

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2 2445	. . 3 |- ( [ x / v ] [ y / u ] [ u / x ] [ v / y ] ph <-> [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph )
2		nfv 1843	. . . . 5 |- F/ u [ v / y ] ph
3	2	sbco2 2415	. . . 4 |- ( [ y / u ] [ u / x ] [ v / y ] ph <-> [ y / x ] [ v / y ] ph )
4	3	sbbii 1887	. . 3 |- ( [ x / v ] [ y / u ] [ u / x ] [ v / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
5	1, 4	bitr3i 266	. 2 |- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
6		sbco4lem 2465	. 2 |- ( [ x / v ] [ y / x ] [ v / y ] ph <-> [ x / t ] [ y / x ] [ t / y ] ph )
7		sbco4lem 2465	. 2 |- ( [ x / t ] [ y / x ] [ t / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )
8	5, 6, 7	3bitri 286	1 |- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )

Syntax hints:   <-> wb 196   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)