Theorem sbcel12g 2014

Description: Distribute proper substitution through a membership relation.

Assertion

Ref	Expression
sbcel12g	|- (A e. D -> ([A / x]B e. C <-> [_A / x]_B e. [_A / x]_C))

Proof of Theorem sbcel12g

Step	Hyp	Ref	Expression
1		elisset 1820	. . 3 |- (A e. D -> A e. V)
2		sbcexg 1978	. . . . 5 |- (A e. V -> ([A / x]E.z(z = B /\ z e. C) <-> E.z[A / x](z = B /\ z e. C)))
3		df-clel 1475	. . . . . 6 |- (B e. C <-> E.z(z = B /\ z e. C))
4	3	sbcbii 1981	. . . . 5 |- (A e. V -> ([A / x]B e. C <-> [A / x]E.z(z = B /\ z e. C)))
5		dfcleq 1473	. . . . . . . . . . 11 |- (z = B <-> A.y(y e. z <-> y e. B))
6	5	sbcbii 1981	. . . . . . . . . 10 |- (A e. V -> ([A / x]z = B <-> [A / x]A.y(y e. z <-> y e. B)))
7		sbcalg 1977	. . . . . . . . . 10 |- (A e. V -> ([A / x]A.y(y e. z <-> y e. B) <-> A.y[A / x](y e. z <-> y e. B)))
8		sbcbidig 1976	. . . . . . . . . . . 12 |- (A e. V -> ([A / x](y e. z <-> y e. B) <-> ([A / x]y e. z <-> [A / x]y e. B)))
9		ax-17 973	. . . . . . . . . . . . . 14 |- (y e. z -> A.x y e. z)
10	9	sbcgf 1989	. . . . . . . . . . . . 13 |- (A e. V -> ([A / x]y e. z <-> y e. z))
11	10	bibi1d 621	. . . . . . . . . . . 12 |- (A e. V -> (([A / x]y e. z <-> [A / x]y e. B) <-> (y e. z <-> [A / x]y e. B)))
12	8, 11	bitrd 530	. . . . . . . . . . 11 |- (A e. V -> ([A / x](y e. z <-> y e. B) <-> (y e. z <-> [A / x]y e. B)))
13	12	albidv 1280	. . . . . . . . . 10 |- (A e. V -> (A.y[A / x](y e. z <-> y e. B) <-> A.y(y e. z <-> [A / x]y e. B)))
14	6, 7, 13	3bitrd 546	. . . . . . . . 9 |- (A e. V -> ([A / x]z = B <-> A.y(y e. z <-> [A / x]y e. B)))
15		abeq2 1571	. . . . . . . . 9 |- (z = {y | [A / x]y e. B} <-> A.y(y e. z <-> [A / x]y e. B))
16	14, 15	syl6rbbr 541	. . . . . . . 8 |- (A e. V -> (z = {y | [A / x]y e. B} <-> [A / x]z = B))
17		eleq1 1537	. . . . . . . . . . . 12 |- (y = z -> (y e. C <-> z e. C))
18	17	sbcbidv 1980	. . . . . . . . . . 11 |- ((y = z /\ A e. V) -> ([A / x]y e. C <-> [A / x]z e. C))
19	18	expcom 374	. . . . . . . . . 10 |- (A e. V -> (y = z -> ([A / x]y e. C <-> [A / x]z e. C)))
20	19	19.21aiv 1288	. . . . . . . . 9 |- (A e. V -> A.y(y = z -> ([A / x]y e. C <-> [A / x]z e. C)))
21		visset 1816	. . . . . . . . . 10 |- z e. V
22		elabgt 1898	. . . . . . . . . 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))
23	21, 22	mpan 697	. . . . . . . . 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))
24	20, 23	syl 10	. . . . . . . 8 |- (A e. V -> (z e. {y | [A / x]y e. C} <-> [A / x]z e. C))
25	16, 24	anbi12d 630	. . . . . . 7 |- (A e. V -> ((z = {y | [A / x]y e. B} /\ z e. {y | [A / x]y e. C}) <-> ([A / x]z = B /\ [A / x]z e. C)))
26		sbcang 1974	. . . . . . 7 |- (A e. V -> ([A / x](z = B /\ z e. C) <-> ([A / x]z = B /\ [A / x]z e. C)))
27	25, 26	bitr4d 533	. . . . . 6 |- (A e. V -> ((z = {y | [A / x]y e. B} /\ z e. {y | [A / x]y e. C}) <-> [A / x](z = B /\ z e. C)))
28	27	exbidv 1281	. . . . 5 |- (A e. V -> (E.z(z = {y | [A / x]y e. B} /\ z e. {y | [A / x]y e. C}) <-> E.z[A / x](z = B /\ z e. C)))
29	2, 4, 28	3bitr4d 552	. . . 4 |- (A e. V -> ([A / x]B e. C <-> E.z(z = {y | [A / x]y e. B} /\ z e. {y | [A / x]y e. C})))
30		df-clel 1475	. . . 4 |- ({y | [A / x]y e. B} e. {y | [A / x]y e. C} <-> E.z(z = {y | [A / x]y e. B} /\ z e. {y | [A / x]y e. C}))
31	29, 30	syl6bbr 540	. . 3 |- (A e. V -> ([A / x]B e. C <-> {y | [A / x]y e. B} e. {y | [A / x]y e. C}))
32	1, 31	syl 10	. 2 |- (A e. D -> ([A / x]B e. C <-> {y | [A / x]y e. B} e. {y | [A / x]y e. C}))
33		df-csb 2005	. . 3 |- [_A / x]_B = {y | [A / x]y e. B}
34		df-csb 2005	. . 3 |- [_A / x]_C = {y | [A / x]y e. C}
35	33, 34	eleq12i 1542	. 2 |- ([_A / x]_B e. [_A / x]_C <-> {y | [A / x]y e. B} e. {y | [A / x]y e. C})
36	32, 35	syl6bbr 540	1 |- (A e. D -> ([A / x]B e. C <-> [_A / x]_B e. [_A / x]_C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172  {cab 1466  Vcvv 1814  [_csb 2004

This theorem is referenced by:  sbcel1g 2016  sbcel2g 2018  sbccsb2g 2026  sbcnestg 2041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945  df-csb 2005

Copyright terms: Public domain