Theorem csbxpg 4814

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 3198	. . 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 3097	. . . . 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 3097	. . . . . . 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 3090	. . . . . . . . 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 3112	. . . . . . . . . 10 ⊢ (A ∈ D → ([̣A / x]̣z = ⟨w, y⟩ ↔ z = ⟨w, y⟩))
6		sbcang 3090	. . . . . . . . . . 11 ⊢ (A ∈ D → ([̣A / x]̣(w ∈ B ∧ y ∈ C) ↔ ([̣A / x]̣w ∈ B ∧ [̣A / x]̣y ∈ C)))
7		sbcel2g 3158	. . . . . . . . . . . 12 ⊢ (A ∈ D → ([̣A / x]̣w ∈ B ↔ w ∈ [A / x]B))
8		sbcel2g 3158	. . . . . . . . . . . 12 ⊢ (A ∈ D → ([̣A / x]̣y ∈ C ↔ y ∈ [A / x]C))
9	7, 8	anbi12d 691	. . . . . . . . . . 11 ⊢ (A ∈ D → (([̣A / x]̣w ∈ B ∧ [̣A / x]̣y ∈ C) ↔ (w ∈ [A / x]B ∧ y ∈ [A / x]C)))
10	6, 9	bitrd 244	. . . . . . . . . 10 ⊢ (A ∈ D → ([̣A / x]̣(w ∈ B ∧ y ∈ C) ↔ (w ∈ [A / x]B ∧ y ∈ [A / x]C)))
11	5, 10	anbi12d 691	. . . . . . . . 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 244	. . . . . . . 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 1626	. . . . . . 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 244	. . . . . 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 1626	. . . . 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 244	. . . 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 2468	. . 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 2385	. 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 4785	. . . 4 ⊢ (B × C) = {⟨w, y⟩ ∣ (w ∈ B ∧ y ∈ C)}
20		df-opab 4624	. . . 4 ⊢ {⟨w, y⟩ ∣ (w ∈ B ∧ y ∈ C)} = {z ∣ ∃w∃y(z = ⟨w, y⟩ ∧ (w ∈ B ∧ y ∈ C))}
21	19, 20	eqtri 2373	. . 3 ⊢ (B × C) = {z ∣ ∃w∃y(z = ⟨w, y⟩ ∧ (w ∈ B ∧ y ∈ C))}
22	21	csbeq2i 3163	. 2 ⊢ [A / x](B × C) = [A / x]{z ∣ ∃w∃y(z = ⟨w, y⟩ ∧ (w ∈ B ∧ y ∈ C))}
23		df-xp 4785	. . 3 ⊢ ([A / x]B × [A / x]C) = {⟨w, y⟩ ∣ (w ∈ [A / x]B ∧ y ∈ [A / x]C)}
24		df-opab 4624	. . 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 2373	. 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 2410	1 ⊢ (A ∈ D → [A / x](B × C) = ([A / x]B × [A / x]C))

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137  ⟨cop 4562  {copab 4623   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138  df-opab 4624  df-xp 4785

This theorem is referenced by:  csbresg  4977