Theorem sbco4lem 2112

Description: Lemma for sbco4 2113. 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 2168. (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 2094	. . 3 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
2	1	sbbidv 2090	. 2 ⊢ (𝑣 = 𝑤 → ([𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑦]𝜑))
3	2	cbvsbv 2111	1 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   ↔ wb 207  [wsb 2073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074

This theorem is referenced by:  sbco4  2113  sbco4OLD  2185