Theorem sbceqdig 2023

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 1824	. . 3 |- (A e. D -> A e. V)
2		sbcalg 1984	. . . . 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 1476	. . . . . 6 |- (B = C <-> A.z(z e. B <-> z e. C))
4	3	sbcbii 1988	. . . . 5 |- (A e. V -> ([A / x]B = C <-> [A / x]A.z(z e. B <-> z e. C)))
5		eleq1 1541	. . . . . . . . . . . 12 |- (y = z -> (y e. B <-> z e. B))
6	5	sbcbidv 1987	. . . . . . . . . . 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 1290	. . . . . . . . 9 |- (A e. V -> A.y(y = z -> ([A / x]y e. B <-> [A / x]z e. B)))
9		visset 1820	. . . . . . . . . 10 |- z e. V
10		elabgt 1902	. . . . . . . . . 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 699	. . . . . . . . 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 1541	. . . . . . . . . . . 12 |- (y = z -> (y e. C <-> z e. C))
14	13	sbcbidv 1987	. . . . . . . . . . 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 1290	. . . . . . . . 9 |- (A e. V -> A.y(y = z -> ([A / x]y e. C <-> [A / x]z e. C)))
17		elabgt 1902	. . . . . . . . . 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 699	. . . . . . . . 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 632	. . . . . . 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 1983	. . . . . . 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 534	. . . . . 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 1282	. . . . 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 553	. . . 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 1476	. . . 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 541	. . 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 2012	. . 3 |- [_A / x]_B = {y | [A / x]y e. B}
29		df-csb 2012	. . 3 |- [_A / x]_C = {y | [A / x]y e. C}
30	28, 29	eqeq12i 1495	. 2 |- ([_A / x]_B = [_A / x]_C <-> {y | [A / x]y e. B} = {y | [A / x]y e. C})
31	27, 30	syl6bbr 541	1 |- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 958   = wceq 960   e. wcel 962  [wsbc 1174  {cab 1469  Vcvv 1818  [_csb 2011

This theorem is referenced by:  sbceq1dig 2025  sbceq2dig 2027  csbeq2d 2029  csbeq2i 2031  fsum1s 7041  fsump1s 7045  csbfsum 7059

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-v 1819  df-sbc 1949  df-csb 2012

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