Theorem csbnestgf 3097

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 2737	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-csb 3046	. . . . . . 7 ⊢ ⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐶}
3	2	abeq2i 2277	. . . . . 6 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐶)
4	3	sbcbii 3010	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶)
5		nfcr 2300	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥 𝑧 ∈ 𝐶)
6	5	alimi 1443	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝐶 → ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶)
7		sbcnestgf 3096	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
8	6, 7	sylan2 284	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
9	4, 8	syl5bb 191	. . . 4 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
10	9	abbidv 2284	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶} = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶})
11	1, 10	sylan 281	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶} = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶})
12		df-csb 3046	. 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶}
13		df-csb 3046	. 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶}
14	11, 12, 13	3eqtr4g 2224	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295  Vcvv 2726  [wsbc 2951  ⦋csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  csbnestg  3099  csbnest1g  3100