Theorem sbco4lemOLD 2207

Description: Obsolete version of sbco4lem 2135 as of 3-Sep-2025. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof shortened by Wolf Lammen, 12-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbco4lemOLD	⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

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

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem sbco4lemOLD

Step	Hyp	Ref	Expression
1		sbcom2 2206	. . 3 ⊢ ([𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
2	1	sbbii 2109	. 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
3		sbco2vv 2133	. . 3 ⊢ ([𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
4	3	2sbbii 2110	. 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
5		sbco2vv 2133	. 2 ⊢ ([𝑥 / 𝑣][𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
6	2, 4, 5	3bitr3i 304	1 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-11 2191

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096

This theorem is used by: (None)