Theorem csbcow 3070

Description: Composition law for chained substitutions into a class. Version of csbco 3069 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 3060	. . . . . 6 |- [_ y / x ]_ B = { z | [. y / x ]. z e. B }
2	1	abeq2i 2288	. . . . 5 |- ( z e. [_ y / x ]_ B <-> [. y / x ]. z e. B )
3	2	sbcbii 3024	. . . 4 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / y ]. [. y / x ]. z e. B )
4		nfv 1528	. . . . . . . . . 10 |- F/ y A. x ( x = w -> z e. B )
5		equequ2 1713	. . . . . . . . . . . 12 |- ( y = w -> ( x = y <-> x = w ) )
6	5	imbi1d 231	. . . . . . . . . . 11 |- ( y = w -> ( ( x = y -> z e. B ) <-> ( x = w -> z e. B ) ) )
7	6	albidv 1824	. . . . . . . . . 10 |- ( y = w -> ( A. x ( x = y -> z e. B ) <-> A. x ( x = w -> z e. B ) ) )
8	4, 7	sbiev 1792	. . . . . . . . 9 |- ( [ w / y ] A. x ( x = y -> z e. B ) <-> A. x ( x = w -> z e. B ) )
9		sb6 1886	. . . . . . . . 9 |- ( [ w / x ] z e. B <-> A. x ( x = w -> z e. B ) )
10	8, 9	bitr4i 187	. . . . . . . 8 |- ( [ w / y ] A. x ( x = y -> z e. B ) <-> [ w / x ] z e. B )
11		df-clab 2164	. . . . . . . 8 |- ( w e. { y | A. x ( x = y -> z e. B ) } <-> [ w / y ] A. x ( x = y -> z e. B ) )
12		df-clab 2164	. . . . . . . 8 |- ( w e. { x | z e. B } <-> [ w / x ] z e. B )
13	10, 11, 12	3bitr4i 212	. . . . . . 7 |- ( w e. { y | A. x ( x = y -> z e. B ) } <-> w e. { x | z e. B } )
14	13	eqriv 2174	. . . . . 6 |- { y | A. x ( x = y -> z e. B ) } = { x | z e. B }
15	14	eleq2i 2244	. . . . 5 |- ( A e. { y | A. x ( x = y -> z e. B ) } <-> A e. { x | z e. B } )
16		df-sbc 2965	. . . . . 6 |- ( [. A / y ]. [. y / x ]. z e. B <-> A e. { y | [. y / x ]. z e. B } )
17		df-sbc 2965	. . . . . . . . 9 |- ( [. y / x ]. z e. B <-> y e. { x | z e. B } )
18		df-clab 2164	. . . . . . . . . 10 |- ( y e. { x | z e. B } <-> [ y / x ] z e. B )
19		sb6 1886	. . . . . . . . . 10 |- ( [ y / x ] z e. B <-> A. x ( x = y -> z e. B ) )
20	18, 19	bitri 184	. . . . . . . . 9 |- ( y e. { x | z e. B } <-> A. x ( x = y -> z e. B ) )
21	17, 20	bitri 184	. . . . . . . 8 |- ( [. y / x ]. z e. B <-> A. x ( x = y -> z e. B ) )
22	21	abbii 2293	. . . . . . 7 |- { y | [. y / x ]. z e. B } = { y | A. x ( x = y -> z e. B ) }
23	22	eleq2i 2244	. . . . . 6 |- ( A e. { y | [. y / x ]. z e. B } <-> A e. { y | A. x ( x = y -> z e. B ) } )
24	16, 23	bitri 184	. . . . 5 |- ( [. A / y ]. [. y / x ]. z e. B <-> A e. { y | A. x ( x = y -> z e. B ) } )
25		df-sbc 2965	. . . . 5 |- ( [. A / x ]. z e. B <-> A e. { x | z e. B } )
26	15, 24, 25	3bitr4i 212	. . . 4 |- ( [. A / y ]. [. y / x ]. z e. B <-> [. A / x ]. z e. B )
27	3, 26	bitri 184	. . 3 |- ( [. A / y ]. z e. [_ y / x ]_ B <-> [. A / x ]. z e. B )
28	27	abbii 2293	. 2 |- { z | [. A / y ]. z e. [_ y / x ]_ B } = { z | [. A / x ]. z e. B }
29		df-csb 3060	. 2 |- [_ A / y ]_ [_ y / x ]_ B = { z | [. A / y ]. z e. [_ y / x ]_ B }
30		df-csb 3060	. 2 |- [_ A / x ]_ B = { z | [. A / x ]. z e. B }
31	28, 29, 30	3eqtr4i 2208	1 |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B

Syntax hints:   -> wi 4   A.wal 1351   = wceq 1353   [wsb 1762   e. wcel 2148   {cab 2163   [.wsbc 2964   [_csb 3059

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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2965  df-csb 3060

This theorem is referenced by:  zproddc  11589  fprodseq  11593