Theorem sbceqdig 2012

Description: Distribute proper substitution through an equality relation.

Assertion

Ref	Expression
sbceqdig	|- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))

Proof of Theorem sbceqdig

Step	Hyp	Ref	Expression
1		elisset 1817	. . 3 |- (A e. D -> A e. V)
2		sbcalg 1974	. . . . 5 |- (A e. V -> ([A / x]A.z(z e. B <-> z e. C) <-> A.z[A / x](z e. B <-> z e. C)))
3		dfcleq 1470	. . . . . 6 |- (B = C <-> A.z(z e. B <-> z e. C))
4	3	sbcbii 1978	. . . . 5 |- (A e. V -> ([A / x]B = C <-> [A / x]A.z(z e. B <-> z e. C)))
5		eleq1 1534	. . . . . . . . . . . 12 |- (y = z -> (y e. B <-> z e. B))
6	5	sbcbidv 1977	. . . . . . . . . . 11 |- ((y = z /\ A e. V) -> ([A / x]y e. B <-> [A / x]z e. B))
7	6	expcom 374	. . . . . . . . . 10 |- (A e. V -> (y = z -> ([A / x]y e. B <-> [A / x]z e. B)))
8	7	19.21aiv 1286	. . . . . . . . 9 |- (A e. V -> A.y(y = z -> ([A / x]y e. B <-> [A / x]z e. B)))
9		visset 1813	. . . . . . . . . 10 |- z e. V
10		elabgt 1895	. . . . . . . . . 10 |- ((z e. V /\ A.y(y = z -> ([A / x]y e. B <-> [A / x]z e. B))) -> (z e. {y | [A / x]y e. B} <-> [A / x]z e. B))
11	9, 10	mpan 695	. . . . . . . . 9 |- (A.y(y = z -> ([A / x]y e. B <-> [A / x]z e. B)) -> (z e. {y | [A / x]y e. B} <-> [A / x]z e. B))
12	8, 11	syl 10	. . . . . . . 8 |- (A e. V -> (z e. {y | [A / x]y e. B} <-> [A / x]z e. B))
13		eleq1 1534	. . . . . . . . . . . 12 |- (y = z -> (y e. C <-> z e. C))
14	13	sbcbidv 1977	. . . . . . . . . . 11 |- ((y = z /\ A e. V) -> ([A / x]y e. C <-> [A / x]z e. C))
15	14	expcom 374	. . . . . . . . . 10 |- (A e. V -> (y = z -> ([A / x]y e. C <-> [A / x]z e. C)))
16	15	19.21aiv 1286	. . . . . . . . 9 |- (A e. V -> A.y(y = z -> ([A / x]y e. C <-> [A / x]z e. C)))
17		elabgt 1895	. . . . . . . . . 10 |- ((z e. V /\ A.y(y = z -> ([A / x]y e. C <-> [A / x]z e. C))) -> (z e. {y | [A / x]y e. C} <-> [A / x]z e. C))
18	9, 17	mpan 695	. . . . . . . . 9 |- (A.y(y = z -> ([A / x]y e. C <-> [A / x]z e. C)) -> (z e. {y | [A / x]y e. C} <-> [A / x]z e. C))
19	16, 18	syl 10	. . . . . . . 8 |- (A e. V -> (z e. {y | [A / x]y e. C} <-> [A / x]z e. C))
20	12, 19	bibi12d 629	. . . . . . 7 |- (A e. V -> ((z e. {y | [A / x]y e. B} <-> z e. {y | [A / x]y e. C}) <-> ([A / x]z e. B <-> [A / x]z e. C)))
21		sbcbidig 1973	. . . . . . 7 |- (A e. V -> ([A / x](z e. B <-> z e. C) <-> ([A / x]z e. B <-> [A / x]z e. C)))
22	20, 21	bitr4d 531	. . . . . 6 |- (A e. V -> ((z e. {y | [A / x]y e. B} <-> z e. {y | [A / x]y e. C}) <-> [A / x](z e. B <-> z e. C)))
23	22	albidv 1278	. . . . 5 |- (A e. V -> (A.z(z e. {y | [A / x]y e. B} <-> z e. {y | [A / x]y e. C}) <-> A.z[A / x](z e. B <-> z e. C)))
24	2, 4, 23	3bitr4d 550	. . . 4 |- (A e. V -> ([A / x]B = C <-> A.z(z e. {y | [A / x]y e. B} <-> z e. {y | [A / x]y e. C})))
25		dfcleq 1470	. . . 4 |- ({y | [A / x]y e. B} = {y | [A / x]y e. C} <-> A.z(z e. {y | [A / x]y e. B} <-> z e. {y | [A / x]y e. C}))
26	24, 25	syl6bbr 538	. . 3 |- (A e. V -> ([A / x]B = C <-> {y | [A / x]y e. B} = {y | [A / x]y e. C}))
27	1, 26	syl 10	. 2 |- (A e. D -> ([A / x]B = C <-> {y | [A / x]y e. B} = {y | [A / x]y e. C}))
28		df-csb 2002	. . 3 |- [_A / x]_B = {y | [A / x]y e. B}
29		df-csb 2002	. . 3 |- [_A / x]_C = {y | [A / x]y e. C}
30	28, 29	eqeq12i 1488	. 2 |- ([_A / x]_B = [_A / x]_C <-> {y | [A / x]y e. B} = {y | [A / x]y e. C})
31	27, 30	syl6bbr 538	1 |- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  [wsbc 1170  {cab 1463  Vcvv 1811  [_csb 2001

This theorem is referenced by:  sbceq1dig 2014  sbceq2dig 2016  csbeq2d 2018  csbeq2i 2020  fsum1s 7009  fsump1s 7013  csbfsum 7027

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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