Theorem sbco4lem 2271

Description: Lemma for sbco4 2273. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof shortened by Wolf Lammen, 12-Oct-2024.)

Assertion

Ref	Expression
sbco4lem	⊢ ([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 sbco4lem

Step	Hyp	Ref	Expression
1		sbcom2 2160	. . 3 ⊢ ([𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
2	1	sbbii 2078	. 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
3		sbco2vv 2099	. . 3 ⊢ ([𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑣 / 𝑦]𝜑)
4	3	2sbbii 2079	. 2 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑)
5		sbco2vv 2099	. 2 ⊢ ([𝑥 / 𝑣][𝑣 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
6	2, 4, 5	3bitr3i 301	1 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-11 2153

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-sb 2067

This theorem is referenced by:  sbco4  2273