Theorem sbco4OLD 2207

Description: Obsolete version of sbco4 2135 as of 3-Sep-2025. (Contributed by Jim Kingdon, 25-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbco4OLD	⊢ ([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 sbco4OLD

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom2 2205	. . 3 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑)
2		sbco2vv 2132	. . . 4 ⊢ ([𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
3	2	sbbii 2108	. . 3 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑢][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
4	1, 3	bitr3i 279	. 2 ⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
5		sbco4lem 2134	. 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑡][𝑦 / 𝑥][𝑡 / 𝑦]𝜑)
6		sbco4lem 2134	. 2 ⊢ ([𝑥 / 𝑡][𝑦 / 𝑥][𝑡 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
7	4, 5, 6	3bitri 299	1 ⊢ ([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 208  [wsb 2089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-11 2190

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-sb 2090

This theorem is referenced by: (None)