Theorem csbing 3463

Description: Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)

Assertion

Ref	Expression
csbing	|- (A e. B -> [_A / x]_(C i^i D) = ([_A / x]_C i^i [_A / x]_D))

Proof of Theorem csbing

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3140	. . 3 |- (y = A -> [_y / x]_(C i^i D) = [_A / x]_(C i^i D))
2		csbeq1 3140	. . . 4 |- (y = A -> [_y / x]_C = [_A / x]_C)
3		csbeq1 3140	. . . 4 |- (y = A -> [_y / x]_D = [_A / x]_D)
4	2, 3	ineq12d 3459	. . 3 |- (y = A -> ([_y / x]_C i^i [_y / x]_D) = ([_A / x]_C i^i [_A / x]_D))
5	1, 4	eqeq12d 2367	. 2 |- (y = A -> ([_y / x]_(C i^i D) = ([_y / x]_C i^i [_y / x]_D) <-> [_A / x]_(C i^i D) = ([_A / x]_C i^i [_A / x]_D)))
6		vex 2863	. . 3 |- y e. _V
7		nfcsb1v 3169	. . . 4 |- F/_x[_y / x]_C
8		nfcsb1v 3169	. . . 4 |- F/_x[_y / x]_D
9	7, 8	nfin 3231	. . 3 |- F/_x([_y / x]_C i^i [_y / x]_D)
10		csbeq1a 3145	. . . 4 |- (x = y -> C = [_y / x]_C)
11		csbeq1a 3145	. . . 4 |- (x = y -> D = [_y / x]_D)
12	10, 11	ineq12d 3459	. . 3 |- (x = y -> (C i^i D) = ([_y / x]_C i^i [_y / x]_D))
13	6, 9, 12	csbief 3178	. 2 |- [_y / x]_(C i^i D) = ([_y / x]_C i^i [_y / x]_D)
14	5, 13	vtoclg 2915	1 |- (A e. B -> [_A / x]_(C i^i D) = ([_A / x]_C i^i [_A / x]_D))

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  [_csb 3137   i^i cin 3209

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-3an 936  df-nan 1288  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-nin 3212  df-compl 3213  df-in 3214

This theorem is referenced by:  csbresg  4977