Theorem csbnestgf 3133

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 2771	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-csb 3081	. . . . . . 7 ⊢ ⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐶}
3	2	abeq2i 2304	. . . . . 6 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐶)
4	3	sbcbii 3045	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶)
5		nfcr 2328	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥 𝑧 ∈ 𝐶)
6	5	alimi 1466	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝐶 → ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶)
7		sbcnestgf 3132	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
8	6, 7	sylan2 286	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
9	4, 8	bitrid 192	. . . 4 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶))
10	9	abbidv 2311	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑦Ⅎ𝑥𝐶) → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶} = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶})
11	1, 10	sylan 283	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶} = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶})
12		df-csb 3081	. 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶}
13		df-csb 3081	. 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶}
14	11, 12, 13	3eqtr4g 2251	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  Ⅎwnf 1471   ∈ wcel 2164  {cab 2179  Ⅎwnfc 2323  Vcvv 2760  [wsbc 2985  ⦋csb 3080

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-csb 3081

This theorem is referenced by:  csbnestg  3135  csbnest1g  3136