Theorem sbco4 2000

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 1980	. . 3 |- ( [ x / v ] [ y / u ] [ u / x ] [ v / y ] ph <-> [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph )
2		nfv 1521	. . . . 5 |- F/ u [ v / y ] ph
3	2	sbco2 1958	. . . 4 |- ( [ y / u ] [ u / x ] [ v / y ] ph <-> [ y / x ] [ v / y ] ph )
4	3	sbbii 1758	. . 3 |- ( [ x / v ] [ y / u ] [ u / x ] [ v / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
5	1, 4	bitr3i 185	. 2 |- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <-> [ x / v ] [ y / x ] [ v / y ] ph )
6		sbco4lem 1999	. 2 |- ( [ x / v ] [ y / x ] [ v / y ] ph <-> [ x / t ] [ y / x ] [ t / y ] ph )
7		sbco4lem 1999	. 2 |- ( [ x / t ] [ y / x ] [ t / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )
8	5, 6, 7	3bitri 205	1 |- ( [ y / u ] [ x / v ] [ u / x ] [ v / y ] ph <-> [ x / w ] [ y / x ] [ w / y ] ph )

Syntax hints:   <-> wb 104   [wsb 1755

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by: (None)