Theorem csbnestgf 3948

Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Assertion

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

Proof of Theorem csbnestgf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3185	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-csb 3500	. . . . . . 7 ⊢ ⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐶}
3	2	abeq2i 2722	. . . . . 6 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐶)
4	3	sbcbii 3458	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶)
5		nfcr 2743	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥 𝑧 ∈ 𝐶)
6	5	alimi 1730	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝐶 → ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶)
7		sbcnestgf 3947	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
8	6, 7	sylan2 490	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
9	4, 8	syl5bb 271	. . . 4 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
10	9	abbidv 2728	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶} = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶})
11	1, 10	sylan 487	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶} = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶})
12		df-csb 3500	. 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶}
13		df-csb 3500	. 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶}
14	11, 12, 13	3eqtr4g 2669	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  Vcvv 3173  [wsbc 3402  ⦋csb 3499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500

This theorem is referenced by:  csbnestg  3950  csbnest1g  3953