Theorem csbcow 3095

Description: Composition law for chained substitutions into a class. Version of csbco 3094 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by GG, 25-Aug-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 variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3085	. . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵}
2	1	abeq2i 2307	. . . . 5 ⊢ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵)
3	2	sbcbii 3049	. . . 4 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵)
4		nfv 1542	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑥(𝑥 = 𝑤 → 𝑧 ∈ 𝐵)
5		equequ2 1727	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑤))
6	5	imbi1d 231	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑥 = 𝑦 → 𝑧 ∈ 𝐵) ↔ (𝑥 = 𝑤 → 𝑧 ∈ 𝐵)))
7	6	albidv 1838	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝑤 → 𝑧 ∈ 𝐵)))
8	4, 7	sbiev 1806	. . . . . . . . 9 ⊢ ([𝑤 / 𝑦]∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝑤 → 𝑧 ∈ 𝐵))
9		sb6 1901	. . . . . . . . 9 ⊢ ([𝑤 / 𝑥]𝑧 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝑤 → 𝑧 ∈ 𝐵))
10	8, 9	bitr4i 187	. . . . . . . 8 ⊢ ([𝑤 / 𝑦]∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵) ↔ [𝑤 / 𝑥]𝑧 ∈ 𝐵)
11		df-clab 2183	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑦 ∣ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵)} ↔ [𝑤 / 𝑦]∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵))
12		df-clab 2183	. . . . . . . 8 ⊢ (𝑤 ∈ {𝑥 ∣ 𝑧 ∈ 𝐵} ↔ [𝑤 / 𝑥]𝑧 ∈ 𝐵)
13	10, 11, 12	3bitr4i 212	. . . . . . 7 ⊢ (𝑤 ∈ {𝑦 ∣ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵)} ↔ 𝑤 ∈ {𝑥 ∣ 𝑧 ∈ 𝐵})
14	13	eqriv 2193	. . . . . 6 ⊢ {𝑦 ∣ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵)} = {𝑥 ∣ 𝑧 ∈ 𝐵}
15	14	eleq2i 2263	. . . . 5 ⊢ (𝐴 ∈ {𝑦 ∣ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵)} ↔ 𝐴 ∈ {𝑥 ∣ 𝑧 ∈ 𝐵})
16		df-sbc 2990	. . . . . 6 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝐴 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵})
17		df-sbc 2990	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝑦 ∈ {𝑥 ∣ 𝑧 ∈ 𝐵})
18		df-clab 2183	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∣ 𝑧 ∈ 𝐵} ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵)
19		sb6 1901	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵))
20	18, 19	bitri 184	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∣ 𝑧 ∈ 𝐵} ↔ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵))
21	17, 20	bitri 184	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵))
22	21	abbii 2312	. . . . . . 7 ⊢ {𝑦 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵} = {𝑦 ∣ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵)}
23	22	eleq2i 2263	. . . . . 6 ⊢ (𝐴 ∈ {𝑦 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵} ↔ 𝐴 ∈ {𝑦 ∣ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵)})
24	16, 23	bitri 184	. . . . 5 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝐴 ∈ {𝑦 ∣ ∀𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝐵)})
25		df-sbc 2990	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝑧 ∈ 𝐵})
26	15, 24, 25	3bitr4i 212	. . . 4 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
27	3, 26	bitri 184	. . 3 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
28	27	abbii 2312	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵} = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
29		df-csb 3085	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵}
30		df-csb 3085	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
31	28, 29, 30	3eqtr4i 2227	1 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:   → wi 4  ∀wal 1362   = wceq 1364  [wsb 1776   ∈ wcel 2167  {cab 2182  [wsbc 2989  ⦋csb 3084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990  df-csb 3085

This theorem is referenced by:  zproddc  11744  fprodseq  11748