Theorem csbnestgf 3111

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 2750	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-csb 3060	. . . . . . 7 ⊢ ⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐶}
3	2	abeq2i 2288	. . . . . 6 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐶)
4	3	sbcbii 3024	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐶)
5		nfcr 2311	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥 𝑧 ∈ 𝐶)
6	5	alimi 1455	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝐶 → ∀𝑦Ⅎ𝑥 𝑧 ∈ 𝐶)
7		sbcnestgf 3110	. . . . . 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 2295	. . 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 3060	. 2 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐶}
13		df-csb 3060	. 2 ⊢ ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶 = {𝑧 ∣ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝑧 ∈ 𝐶}
14	11, 12, 13	3eqtr4g 2235	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  Ⅎwnf 1460   ∈ wcel 2148  {cab 2163  Ⅎwnfc 2306  Vcvv 2739  [wsbc 2964  ⦋csb 3059

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-csb 3060

This theorem is referenced by:  csbnestg  3113  csbnest1g  3114