Theorem csbcow 3038

Description: Composition law for chained substitutions into a class. Version of csbco 3037 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 25-Aug-2024.)

Assertion

Ref	Expression
csbcow	|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Distinct variable groups:   x, y   y, B

Allowed substitution hints:   A( x, y)   B( x)

Proof of Theorem csbcow

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

Step	Hyp	Ref	Expression
1		df-csb 3028	. . . . . 6 |- [_ y / x ]_ B = { z | [. y / x ]. z e. B }
2	1	abeq2i 2265	. . . . 5 |- ( z e. [_ y / x ]_ B <-> [. y / x ]. z e. B )
3	2	sbcbii 2992	. . . 4 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / y ]. [. y / x ]. z e. B )
4		nfv 1505	. . . . . . . . . 10 |- F/ y A. x ( x = w -> z e. B )
5		equequ2 1690	. . . . . . . . . . . 12 |- ( y = w -> ( x = y <-> x = w ) )
6	5	imbi1d 230	. . . . . . . . . . 11 |- ( y = w -> ( ( x = y -> z e. B ) <-> ( x = w -> z e. B ) ) )
7	6	albidv 1801	. . . . . . . . . 10 |- ( y = w -> ( A. x ( x = y -> z e. B ) <-> A. x ( x = w -> z e. B ) ) )
8	4, 7	sbiev 1769	. . . . . . . . 9 |- ( [ w / y ] A. x ( x = y -> z e. B ) <-> A. x ( x = w -> z e. B ) )
9		sb6 1863	. . . . . . . . 9 |- ( [ w / x ] z e. B <-> A. x ( x = w -> z e. B ) )
10	8, 9	bitr4i 186	. . . . . . . 8 |- ( [ w / y ] A. x ( x = y -> z e. B ) <-> [ w / x ] z e. B )
11		df-clab 2141	. . . . . . . 8 |- ( w e. { y | A. x ( x = y -> z e. B ) } <-> [ w / y ] A. x ( x = y -> z e. B ) )
12		df-clab 2141	. . . . . . . 8 |- ( w e. { x | z e. B } <-> [ w / x ] z e. B )
13	10, 11, 12	3bitr4i 211	. . . . . . 7 |- ( w e. { y | A. x ( x = y -> z e. B ) } <-> w e. { x | z e. B } )
14	13	eqriv 2151	. . . . . 6 |- { y | A. x ( x = y -> z e. B ) } = { x | z e. B }
15	14	eleq2i 2221	. . . . 5 |- ( A e. { y | A. x ( x = y -> z e. B ) } <-> A e. { x | z e. B } )
16		df-sbc 2934	. . . . . 6 |- ( [. A / y ]. [. y / x ]. z e. B <-> A e. { y | [. y / x ]. z e. B } )
17		df-sbc 2934	. . . . . . . . 9 |- ( [. y / x ]. z e. B <-> y e. { x | z e. B } )
18		df-clab 2141	. . . . . . . . . 10 |- ( y e. { x | z e. B } <-> [ y / x ] z e. B )
19		sb6 1863	. . . . . . . . . 10 |- ( [ y / x ] z e. B <-> A. x ( x = y -> z e. B ) )
20	18, 19	bitri 183	. . . . . . . . 9 |- ( y e. { x | z e. B } <-> A. x ( x = y -> z e. B ) )
21	17, 20	bitri 183	. . . . . . . 8 |- ( [. y / x ]. z e. B <-> A. x ( x = y -> z e. B ) )
22	21	abbii 2270	. . . . . . 7 |- { y | [. y / x ]. z e. B } = { y | A. x ( x = y -> z e. B ) }
23	22	eleq2i 2221	. . . . . 6 |- ( A e. { y | [. y / x ]. z e. B } <-> A e. { y | A. x ( x = y -> z e. B ) } )
24	16, 23	bitri 183	. . . . 5 |- ( [. A / y ]. [. y / x ]. z e. B <-> A e. { y | A. x ( x = y -> z e. B ) } )
25		df-sbc 2934	. . . . 5 |- ( [. A / x ]. z e. B <-> A e. { x | z e. B } )
26	15, 24, 25	3bitr4i 211	. . . 4 |- ( [. A / y ]. [. y / x ]. z e. B <-> [. A / x ]. z e. B )
27	3, 26	bitri 183	. . 3 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / x ]. z e. B )
28	27	abbii 2270	. 2 |- { z | [. A / y ]. z e. [_ y / x ]_ B } = { z | [. A / x ]. z e. B }
29		df-csb 3028	. 2 |- [_ A / y ]_ [_ y / x ]_ B = { z | [. A / y ]. z e. [_ y / x ]_ B }
30		df-csb 3028	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
31	28, 29, 30	3eqtr4i 2185	1 |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Syntax hints:   -> wi 4   A.wal 1330   = wceq 1332   [wsb 1739   e. wcel 2125   {cab 2140   [.wsbc 2933   [_csb 3027

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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-sbc 2934  df-csb 3028

This theorem is referenced by:  zproddc  11453  fprodseq  11457