Theorem sbco4 2007

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 1987	. . 3 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
2		nfv 1528	. . . . 5 ⊢ Ⅎ𝑢[𝑣 / 𝑦]𝜑
3	2	sbco2 1965	. . . 4 ⊢ ([𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
4	3	sbbii 1765	. . 3 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
5	1, 4	bitr3i 186	. 2 ⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
6		sbco4lem 2006	. 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑡][𝑦 / 𝑥][𝑡 / 𝑦]𝜑)
7		sbco4lem 2006	. 2 ⊢ ([𝑥 / 𝑡][𝑦 / 𝑥][𝑡 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
8	5, 6, 7	3bitri 206	1 ⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 105  [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by: (None)