Theorem sbco2vv 2107

Description: A composition law for substitution. Version of sbco2 2552 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 2089	. 2 ⊢ (𝑧 = 𝑤 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑))
2		sbequ 2089	. 2 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	sbievw2 2106	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 208  [wsb 2068

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 1969  ax-7 2014

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069

This theorem is referenced by:  sbco4lem  2282  sbco4  2283  cbvabv  2888  clelsb3vOLD  2940  sbralie  3468  sbccow  3791  wl-equsb3  34827  wl-dfrabv  34897  wl-dfrabf  34899  2reu8i  43386