Theorem sbco2vv 2099

Description: A composition law for substitution. Version of sbco2 2519 with disjoint variable conditions and fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 29-Apr-2023.)

Assertion

Ref	Expression
sbco2vv	⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑧   𝜑,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbco2vv

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbequ 2083	. 2 ⊢ (𝑧 = 𝑤 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑))
2		sbequ 2083	. 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	sbievw2 2098	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065

This theorem is referenced by:  cbvsbv  2100  sbco4lemOLD  2175  sbco4OLD  2176  sbco4lemOLDOLD  2282  sbralie  3366  sbralieALT  3367  sbccow  3827  wl-equsb3  37510  2reu8i  47028