Theorem csbcow 3865

Description: Composition law for chained substitutions into a class. Version of csbco 3866 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 10-Nov-2005.) Avoid ax-13 2402. (Revised by GG, 10-Jan-2024.)

Assertion

Ref	Expression
csbcow	⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Distinct variable groups:   𝑥,𝑦   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem csbcow

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3851	. . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵}
2	1	eqabri 2903	. . . . 5 ⊢ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵)
3	2	sbcbii 3798	. . . 4 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵)
4		sbccow 3765	. . . 4 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
5	3, 4	bitri 277	. . 3 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
6	5	abbii 2828	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵} = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
7		df-csb 3851	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵}
8		df-csb 3851	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
9	6, 7, 8	3eqtr4i 2794	1 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:   = wceq 1559   ∈ wcel 2141  {cab 2739  [wsbc 3742  ⦋csb 3850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-sbc 3743  df-csb 3851

This theorem is referenced by:  csbnest1g  4383  csbvarg  4385  fvmpocurryd  8244  zsum  15735  fsum  15737  fsumsplitf  15759  zprod  15957  fprod  15961  gsumply1eq  22359  f1od2  32881  bj-csbsn  37349  sbccom2  38584  disjinfi  45730  climinf2mpt  46248  climinfmpt  46249  dvmptmulf  46471