Theorem sbceqdig 1983

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 1792	. . 3 |- (A e. D -> A e. V)
2		sbcalg 1945	. . . . 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 1447	. . . . . 6 |- (B = C <-> A.z(z e. B <-> z e. C))
4	3	sbcbii 1949	. . . . 5 |- (A e. V -> ([A / x]B = C <-> [A / x]A.z(z e. B <-> z e. C)))
5		eleq1 1510	. . . . . . . . . . . 12 |- (y = z -> (y e. B <-> z e. B))
6	5	sbcbidv 1948	. . . . . . . . . . 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 1268	. . . . . . . . 9 |- (A e. V -> A.y(y = z -> ([A / x]y e. B <-> [A / x]z e. B)))
9		visset 1788	. . . . . . . . . 10 |- z e. V
10		elabgt 1867	. . . . . . . . . 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 692	. . . . . . . . 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 1510	. . . . . . . . . . . 12 |- (y = z -> (y e. C <-> z e. C))
14	13	sbcbidv 1948	. . . . . . . . . . 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 1268	. . . . . . . . 9 |- (A e. V -> A.y(y = z -> ([A / x]y e. C <-> [A / x]z e. C)))
17		elabgt 1867	. . . . . . . . . 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 692	. . . . . . . . 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 627	. . . . . . 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 1944	. . . . . . 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 529	. . . . . 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 1260	. . . . 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 548	. . . 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 1447	. . . 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 536	. . 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 1973	. . 3 |- [_A / x]_B = {y | [A / x]y e. B}
29		df-csb 1973	. . 3 |- [_A / x]_C = {y | [A / x]y e. C}
30	28, 29	eqeq12i 1464	. 2 |- ([_A / x]_B = [_A / x]_C <-> {y | [A / x]y e. B} = {y | [A / x]y e. C})
31	27, 30	syl6bbr 536	1 |- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 950   = wceq 1099   e. wcel 1105  [wsbc 1153  {cab 1440  Vcvv 1786  [_csb 1972

This theorem is referenced by:  sbceq1dig 1985  sbceq2dig 1987  csbeq2d 1989  csbeq2i 1991  fsum1s 6898  fsump1s 6902  csbfsum 6916

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-sbc 1913  df-csb 1973

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