Theorem sbceqdigOLD 2555

Description: Distribute proper substitution through an equality relation.

Assertion

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

Proof of Theorem sbceqdigOLD

Step	Hyp	Ref	Expression
1		elisset 2299	. . 3 |- (A e. D -> A e. _V)
2		sbcalg 2500	. . . . 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 1878	. . . . . 6 |- (B = C <-> A.z(z e. B <-> z e. C))
4	3	sbcbii 2506	. . . . 5 |- (A e. _V -> ([A / x]B = C <-> [A / x]A.z(z e. B <-> z e. C)))
5		elabgt 2400	. . . . . . . . 9 |- ((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))
6		visset 2295	. . . . . . . . 9 |- z e. _V
7		eleq1 1957	. . . . . . . . . . . 12 |- (y = z -> (y e. B <-> z e. B))
8	7	sbcbidv 2505	. . . . . . . . . . 11 |- ((y = z /\ A e. _V) -> ([A / x]y e. B <-> [A / x]z e. B))
9	8	expcom 403	. . . . . . . . . 10 |- (A e. _V -> (y = z -> ([A / x]y e. B <-> [A / x]z e. B)))
10	9	19.21aiv 1664	. . . . . . . . 9 |- (A e. _V -> A.y(y = z -> ([A / x]y e. B <-> [A / x]z e. B)))
11	5, 6, 10	sylancr 526	. . . . . . . 8 |- (A e. _V -> (z e. {y | [A / x]y e. B} <-> [A / x]z e. B))
12		elabgt 2400	. . . . . . . . 9 |- ((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))
13		eleq1 1957	. . . . . . . . . . . 12 |- (y = z -> (y e. C <-> z e. C))
14	13	sbcbidv 2505	. . . . . . . . . . 11 |- ((y = z /\ A e. _V) -> ([A / x]y e. C <-> [A / x]z e. C))
15	14	expcom 403	. . . . . . . . . 10 |- (A e. _V -> (y = z -> ([A / x]y e. C <-> [A / x]z e. C)))
16	15	19.21aiv 1664	. . . . . . . . 9 |- (A e. _V -> A.y(y = z -> ([A / x]y e. C <-> [A / x]z e. C)))
17	12, 6, 16	sylancr 526	. . . . . . . 8 |- (A e. _V -> (z e. {y | [A / x]y e. C} <-> [A / x]z e. C))
18	11, 17	bibi12d 691	. . . . . . 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)))
19		sbcbidig 2499	. . . . . . 7 |- (A e. _V -> ([A / x](z e. B <-> z e. C) <-> ([A / x]z e. B <-> [A / x]z e. C)))
20	18, 19	bitr4d 590	. . . . . 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)))
21	20	albidv 1656	. . . . 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)))
22	2, 4, 21	3bitr4d 609	. . . 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})))
23		dfcleq 1878	. . . 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}))
24	22, 23	syl6bbr 597	. . 3 |- (A e. _V -> ([A / x]B = C <-> {y | [A / x]y e. B} = {y | [A / x]y e. C}))
25	1, 24	syl 12	. 2 |- (A e. D -> ([A / x]B = C <-> {y | [A / x]y e. B} = {y | [A / x]y e. C}))
26		df-csb 2541	. . 3 |- [_A / x]_B = {y | [A / x]y e. B}
27		df-csb 2541	. . 3 |- [_A / x]_C = {y | [A / x]y e. C}
28	26, 27	eqeq12i 1897	. 2 |- ([_A / x]_B = [_A / x]_C <-> {y | [A / x]y e. B} = {y | [A / x]y e. C})
29	25, 28	syl6bbr 597	1 |- (A e. D -> ([A / x]B = C <-> [_A / x]_B = [_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  [wsbc 1534  {cab 1871  _Vcvv 2292  [_csb 2540

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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