Theorem sbco4lem 2102

Description: Lemma for sbco4 2103. 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.) Avoid ax-11 2158. (Revised by SN, 3-Sep-2025.)

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		sbequ 2084	. . 3 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
2	1	sbbidv 2080	. 2 ⊢ (𝑣 = 𝑤 → ([𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑦]𝜑))
3	2	cbvsbv 2101	1 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 206  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066

This theorem is referenced by:  sbco4  2103  sbco4OLD  2176