Theorem sbco4 1980

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]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbco4

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2 1960	. . 3 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
2		nfv 1508	. . . . 5 ⊢ Ⅎ𝑢[𝑣 / 𝑦]𝜑
3	2	sbco2 1936	. . . 4 ⊢ ([𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
4	3	sbbii 1738	. . 3 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
5	1, 4	bitr3i 185	. 2 ⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
6		sbco4lem 1979	. 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑡][𝑦 / 𝑥][𝑡 / 𝑦]𝜑)
7		sbco4lem 1979	. 2 ⊢ ([𝑥 / 𝑡][𝑦 / 𝑥][𝑡 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
8	5, 6, 7	3bitri 205	1 ⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 104  [wsb 1735

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by: (None)