Theorem sbco2vv 2135

Description: A composition law for substitution. Version of sbco2 2540 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 2119	. 2 ⊢ (𝑧 = 𝑤 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑))
2		sbequ 2119	. 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	sbievw2 2134	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099

This theorem is used by:  cbvsbv  2136  sbralie  3338  sbralieALT  3339  sbralieOLD  3340  sbccow  3761  wl-equsb3  38595  2reu8i  48290