Theorem sbco4lem 2139

Description: Lemma for sbco4 2140. 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 2195. (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 2120	. . 3 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
2	1	sbbidv 2116	. 2 ⊢ (𝑣 = 𝑤 → ([𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑦]𝜑))
3	2	cbvsbv 2138	1 ⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  sbco4  2140  sbco4OLD  2212