Theorem sbco2v 2331

Description: A composition law for substitution. Version of sbco2 2515 with disjoint variable conditions but not requiring ax-13 2372. (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 2328	. 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
3		sbequ 2087	. 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	2, 3	sbiev 2312	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  Ⅎwnf 1787  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-sb 2069

This theorem is referenced by:  cbvabwOLD  2814  clelsb1fw  2910  ichbi12i  44800