Theorem sbco4 2280

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 2165	. . 3 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
2		sbco2vv 2105	. . . 4 ⊢ ([𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
3	2	sbbii 2081	. . 3 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
4	1, 3	bitr3i 280	. 2 ⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
5		sbco4lem 2279	. 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑡][𝑦 / 𝑥][𝑡 / 𝑦]𝜑)
6		sbco4lem 2279	. 2 ⊢ ([𝑥 / 𝑡][𝑦 / 𝑥][𝑡 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
7	4, 5, 6	3bitri 300	1 ⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 209  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070

This theorem is referenced by:  dfich2  44343