Theorem sbco2v 2366

Description: A composition law for substitution. Version of sbco2 2545 with disjoint variable conditions but not requiring ax-13 2406. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 29-Apr-2023.)

Hypothesis

Ref	Expression
sbco2v.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
sbco2v	⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem sbco2v

Step	Hyp	Ref	Expression
1		sbco2v.1	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	nfsbv 2365	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
3		sbequ 2120	. 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	sbiev 2349	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  Ⅎwnf 1816  [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  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  clelsb1fw  2931  ichbi12i  48269