Theorem csbxpg 4685

Description: Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
csbxpg	⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem csbxpg

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbabg 3106	. . 3 ⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))})
2		sbcexg 3005	. . . . 5 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
3		sbcexg 3005	. . . . . . 7 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
4		sbcang 2994	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
5		sbcg 3020	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩))
6		sbcang 2994	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
7		sbcel2g 3066	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵))
8		sbcel2g 3066	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
9	7, 8	anbi12d 465	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐷 → (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
10	6, 9	bitrd 187	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
11	5, 10	anbi12d 465	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → (([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ∧ [𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
12	4, 11	bitrd 187	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
13	12	exbidv 1813	. . . . . . 7 ⊢ (𝐴 ∈ 𝐷 → (∃𝑦[𝐴 / 𝑥](𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
14	3, 13	bitrd 187	. . . . . 6 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
15	14	exbidv 1813	. . . . 5 ⊢ (𝐴 ∈ 𝐷 → (∃𝑤[𝐴 / 𝑥]∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
16	2, 15	bitrd 187	. . . 4 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))))
17	16	abbidv 2284	. . 3 ⊢ (𝐴 ∈ 𝐷 → {𝑧 ∣ [𝐴 / 𝑥]∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))})
18	1, 17	eqtrd 2198	. 2 ⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))})
19		df-xp 4610	. . . 4 ⊢ (𝐵 × 𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
20		df-opab 4044	. . . 4 ⊢ {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
21	19, 20	eqtri 2186	. . 3 ⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
22	21	csbeq2i 3072	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
23		df-xp 4610	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)}
24		df-opab 4044	. . 3 ⊢ {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
25	23, 24	eqtri 2186	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))}
26	18, 22, 25	3eqtr4g 2224	1 ⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  [wsbc 2951  ⦋csb 3045  ⟨cop 3579  {copab 4042   × cxp 4602

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-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  df-opab 4044  df-xp 4610

This theorem is referenced by:  csbresg  4887