Theorem csbnestgfw 4386

Description: Nest the composition of two substitutions. Version of csbnestgf 4391 with a disjoint variable condition, which does not require ax-13 2403. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2403. (Revised by GG, 26-Jan-2024.)

Assertion

Ref	Expression
csbnestgfw	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem csbnestgfw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3475	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-csb 3853	. . . . . . 7 ⊢ ⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐶}
3	2	eqabri 2904	. . . . . 6 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐶)
4	3	sbcbii 3799	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶)
5		nfcr 2914	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥 𝑧 ∈ 𝐶)
6	5	alimi 1840	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝐶 → ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶)
7		sbcnestgfw 4385	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
8	6, 7	sylan2 604	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
9	4, 8	bitrid 286	. . . 4 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
10	9	abbidv 2828	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶} = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶})
11	1, 10	sylan 591	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶} = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶})
12		df-csb 3853	. 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶}
13		df-csb 3853	. 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶}
14	11, 12, 13	3eqtr4g 2822	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1567   = wceq 1569  Ⅎwnf 1812   ∈ wcel 2142  {cab 2740  Ⅎwnfc 2909  Vcvv 3454  [wsbc 3743  ⦋csb 3852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-v 3456  df-sbc 3744  df-csb 3853

This theorem is used by:  csbnestgw  4388  csbnest1g  4396