Theorem csbxpg 4813

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

Assertion

Ref	Expression
csbxpg	|- (A e. D -> [_A / x]_(B X. C) = ([_A / x]_B X. [_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 3197	. . 3 |- (A e. D -> [_A / x]_{z | E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C))} = {z | [.A / x ].E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C))})
2		sbcexg 3096	. . . . 5 |- (A e. D -> ( [.A / x ].E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C)) <-> E.w [.A / x ].E.y(z = <.w, y>. /\ (w e. B /\ y e. C))))
3		sbcexg 3096	. . . . . . 7 |- (A e. D -> ( [.A / x ].E.y(z = <.w, y>. /\ (w e. B /\ y e. C)) <-> E.y [.A / x ].(z = <.w, y>. /\ (w e. B /\ y e. C))))
4		sbcang 3089	. . . . . . . . 9 |- (A e. D -> ( [.A / x ].(z = <.w, y>. /\ (w e. B /\ y e. C)) <-> ( [.A / x ].z = <.w, y>. /\ [.A / x ].(w e. B /\ y e. C))))
5		sbcg 3111	. . . . . . . . . 10 |- (A e. D -> ( [.A / x ].z = <.w, y>. <-> z = <.w, y>.))
6		sbcang 3089	. . . . . . . . . . 11 |- (A e. D -> ( [.A / x ].(w e. B /\ y e. C) <-> ( [.A / x ].w e. B /\ [.A / x ].y e. C)))
7		sbcel2g 3157	. . . . . . . . . . . 12 |- (A e. D -> ( [.A / x ].w e. B <-> w e. [_A / x]_B))
8		sbcel2g 3157	. . . . . . . . . . . 12 |- (A e. D -> ( [.A / x ].y e. C <-> y e. [_A / x]_C))
9	7, 8	anbi12d 691	. . . . . . . . . . 11 |- (A e. D -> (( [.A / x ].w e. B /\ [.A / x ].y e. C) <-> (w e. [_A / x]_B /\ y e. [_A / x]_C)))
10	6, 9	bitrd 244	. . . . . . . . . 10 |- (A e. D -> ( [.A / x ].(w e. B /\ y e. C) <-> (w e. [_A / x]_B /\ y e. [_A / x]_C)))
11	5, 10	anbi12d 691	. . . . . . . . 9 |- (A e. D -> (( [.A / x ].z = <.w, y>. /\ [.A / x ].(w e. B /\ y e. C)) <-> (z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))))
12	4, 11	bitrd 244	. . . . . . . 8 |- (A e. D -> ( [.A / x ].(z = <.w, y>. /\ (w e. B /\ y e. C)) <-> (z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))))
13	12	exbidv 1626	. . . . . . 7 |- (A e. D -> (E.y [.A / x ].(z = <.w, y>. /\ (w e. B /\ y e. C)) <-> E.y(z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))))
14	3, 13	bitrd 244	. . . . . 6 |- (A e. D -> ( [.A / x ].E.y(z = <.w, y>. /\ (w e. B /\ y e. C)) <-> E.y(z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))))
15	14	exbidv 1626	. . . . 5 |- (A e. D -> (E.w [.A / x ].E.y(z = <.w, y>. /\ (w e. B /\ y e. C)) <-> E.wE.y(z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))))
16	2, 15	bitrd 244	. . . 4 |- (A e. D -> ( [.A / x ].E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C)) <-> E.wE.y(z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))))
17	16	abbidv 2467	. . 3 |- (A e. D -> {z | [.A / x ].E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C))} = {z | E.wE.y(z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))})
18	1, 17	eqtrd 2385	. 2 |- (A e. D -> [_A / x]_{z | E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C))} = {z | E.wE.y(z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))})
19		df-xp 4784	. . . 4 |- (B X. C) = {<.w, y>. | (w e. B /\ y e. C)}
20		df-opab 4623	. . . 4 |- {<.w, y>. | (w e. B /\ y e. C)} = {z | E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C))}
21	19, 20	eqtri 2373	. . 3 |- (B X. C) = {z | E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C))}
22	21	csbeq2i 3162	. 2 |- [_A / x]_(B X. C) = [_A / x]_{z | E.wE.y(z = <.w, y>. /\ (w e. B /\ y e. C))}
23		df-xp 4784	. . 3 |- ([_A / x]_B X. [_A / x]_C) = {<.w, y>. | (w e. [_A / x]_B /\ y e. [_A / x]_C)}
24		df-opab 4623	. . 3 |- {<.w, y>. | (w e. [_A / x]_B /\ y e. [_A / x]_C)} = {z | E.wE.y(z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))}
25	23, 24	eqtri 2373	. 2 |- ([_A / x]_B X. [_A / x]_C) = {z | E.wE.y(z = <.w, y>. /\ (w e. [_A / x]_B /\ y e. [_A / x]_C))}
26	18, 22, 25	3eqtr4g 2410	1 |- (A e. D -> [_A / x]_(B X. C) = ([_A / x]_B X. [_A / x]_C))

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   [.wsbc 3046  [_csb 3136  <.cop 4561  {copab 4622   X. cxp 4770

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 2478  df-v 2861  df-sbc 3047  df-csb 3137  df-opab 4623  df-xp 4784

This theorem is referenced by:  csbresg  4976