Theorem elrabsf 1456

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 1421 has implicit substitution). The hypothesis specifies that x must not be a free variable in B.

Hypothesis

Ref	Expression
elrabsf.1	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
elrabsf	|- (A e. {x e. B | ph} <-> (A e. B /\ [A / x]ph))

Distinct variable group(s):   y,B   x,y

Proof of Theorem elrabsf

Step	Hyp	Ref	Expression
1		elrabsf.1	. . . 4 |- (y e. B -> A.x y e. B)
2		ax-17 925	. . . 4 |- (y e. B -> A.z y e. B)
3		ax-17 925	. . . 4 |- (ph -> A.zph)
4		hbs1 986	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
5		sbequ12 865	. . . 4 |- (x = z -> (ph <-> [z / x]ph))
6	1, 2, 3, 4, 5	cbvrab 1425	. . 3 |- {x e. B | ph} = {z e. B | [z / x]ph}
7	6	eleq2i 1153	. 2 |- (A e. {x e. B | ph} <-> A e. {z e. B | [z / x]ph})
8		ax-17 925	. . . 4 |- (w e. A -> A.z w e. A)
9		ax-17 925	. . . 4 |- (w e. B -> A.z w e. B)
10	8	hbsbc 1446	. . . 4 |- ((A e. V -> [A / z][z / x]ph) -> A.z(A e. V -> [A / z][z / x]ph))
11		sbceq1 1443	. . . . 5 |- (z = A -> ([z / x]ph <-> [A / z][z / x]ph))
12		19.8a 712	. . . . . . 7 |- (z = A -> E.z z = A)
13		isset 1351	. . . . . . 7 |- (A e. V <-> E.z z = A)
14	12, 13	sylibr 175	. . . . . 6 |- (z = A -> A e. V)
15		biimt 549	. . . . . 6 |- (A e. V -> ([A / z][z / x]ph <-> (A e. V -> [A / z][z / x]ph)))
16	14, 15	syl 12	. . . . 5 |- (z = A -> ([A / z][z / x]ph <-> (A e. V -> [A / z][z / x]ph)))
17	11, 16	bitrd 406	. . . 4 |- (z = A -> ([z / x]ph <-> (A e. V -> [A / z][z / x]ph)))
18	8, 9, 10, 17	elrabf 1421	. . 3 |- (A e. {z e. B | [z / x]ph} <-> (A e. B /\ (A e. V -> [A / z][z / x]ph)))
19		elisset 1354	. . . . 5 |- (A e. B -> A e. V)
20	19, 15	syl 12	. . . 4 |- (A e. B -> ([A / z][z / x]ph <-> (A e. V -> [A / z][z / x]ph)))
21	20	pm5.32i 489	. . 3 |- ((A e. B /\ [A / z][z / x]ph) <-> (A e. B /\ (A e. V -> [A / z][z / x]ph)))
22	18, 21	bitr4 154	. 2 |- (A e. {z e. B | [z / x]ph} <-> (A e. B /\ [A / z][z / x]ph))
23		sbcco 1448	. . 3 |- (A e. B -> ([A / z][z / x]ph <-> [A / x]ph))
24	23	pm5.32i 489	. 2 |- ((A e. B /\ [A / z][z / x]ph) <-> (A e. B /\ [A / x]ph))
25	7, 22, 24	3bitr 155	1 |- (A e. {x e. B | ph} <-> (A e. B /\ [A / x]ph))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678  [wsb 852   = wceq 1091   e. wcel 1092  {crab 1204  Vcvv 1348  [wsbc 1440

This theorem is referenced by:  elabs2 1457  iunrab 2022  tfis 2245  onminesb 2265

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-v 1349  df-sbc 1441

metamath.org