Theorem csbxpg 4807

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 3189	. . 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 3086	. . . . 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 3086	. . . . . . 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 3075	. . . . . . . . 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 3101	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩))
6		sbcang 3075	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶)))
7		sbcel2g 3148	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝑤 ∈ 𝐵 ↔ 𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵))
8		sbcel2g 3148	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶))
9	7, 8	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐷 → (([𝐴 / 𝑥]𝑤 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
10	6, 9	bitrd 188	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥](𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)))
11	5, 10	anbi12d 473	. . . . . . . . 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 188	. . . . . . . 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 1873	. . . . . . 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 188	. . . . . 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 1873	. . . . 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 188	. . . 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 2349	. . 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 2264	. 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 4731	. . . 4 ⊢ (𝐵 × 𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}
20		df-opab 4151	. . . 4 ⊢ {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
21	19, 20	eqtri 2252	. . 3 ⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
22	21	csbeq2i 3154	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = ⦋𝐴 / 𝑥⦌{𝑧 ∣ ∃𝑤∃𝑦(𝑧 = ⟨𝑤, 𝑦⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))}
23		df-xp 4731	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶) = {⟨𝑤, 𝑦⟩ ∣ (𝑤 ∈ ⦋𝐴 / 𝑥⦌𝐵 ∧ 𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐶)}
24		df-opab 4151	. . 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 2252	. 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 2289	1 ⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217  [wsbc 3031  ⦋csb 3127  ⟨cop 3672  {copab 4149   × cxp 4723

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032  df-csb 3128  df-opab 4151  df-xp 4731

This theorem is referenced by:  csbresg  5016