Theorem elrabsf 2486

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2413 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:   y,B

Proof of Theorem elrabsf

Step	Hyp	Ref	Expression
1		elrabsf.1	. . . 4 |- (y e. B -> A.x y e. B)
2		ax-17 1317	. . . 4 |- (y e. B -> A.z y e. B)
3		ax-17 1317	. . . 4 |- (ph -> A.zph)
4		hbs1 1722	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
5		sbequ12 1545	. . . 4 |- (x = z -> (ph <-> [z / x]ph))
6	1, 2, 3, 4, 5	cbvrab 2421	. . 3 |- {x e. B | ph} = {z e. B | [z / x]ph}
7	6	eleq2i 1961	. 2 |- (A e. {x e. B | ph} <-> A e. {z e. B | [z / x]ph})
8		ax-17 1317	. . . 4 |- (w e. A -> A.z w e. A)
9		ax-17 1317	. . . 4 |- (w e. B -> A.z w e. B)
10	8	hbsbc1 2462	. . . 4 |- ((A e. _V -> [A / z][z / x]ph) -> A.z(A e. _V -> [A / z][z / x]ph))
11		sbceq1a 2456	. . . . 5 |- (z = A -> ([z / x]ph <-> [A / z][z / x]ph))
12		19.8a 1376	. . . . . . 7 |- (z = A -> E.z z = A)
13		isset 2296	. . . . . . 7 |- (A e. _V <-> E.z z = A)
14	12, 13	sylibr 217	. . . . . 6 |- (z = A -> A e. _V)
15		biimt 803	. . . . . 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 587	. . . 4 |- (z = A -> ([z / x]ph <-> (A e. _V -> [A / z][z / x]ph)))
18	8, 9, 10, 17	elrabf 2413	. . 3 |- (A e. {z e. B | [z / x]ph} <-> (A e. B /\ (A e. _V -> [A / z][z / x]ph)))
19		elisset 2299	. . . . 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 707	. . 3 |- ((A e. B /\ [A / z][z / x]ph) <-> (A e. B /\ (A e. _V -> [A / z][z / x]ph)))
22	18, 21	bitr4i 193	. 2 |- (A e. {z e. B | [z / x]ph} <-> (A e. B /\ [A / z][z / x]ph))
23		sbccog 2467	. . 3 |- (A e. B -> ([A / z][z / x]ph <-> [A / x]ph))
24	23	pm5.32i 707	. 2 |- ((A e. B /\ [A / z][z / x]ph) <-> (A e. B /\ [A / x]ph))
25	7, 22, 24	3bitri 194	1 |- (A e. {x e. B | ph} <-> (A e. B /\ [A / x]ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {crab 2108  _Vcvv 2292

This theorem is referenced by:  elabs2 2487  iunrab 3299  reucl2 3814  onminesb 3880  tfis 3938  riotacl2 5578  tfisg 13912  wfisg 13917  frinsg 13941

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-rab 2112  df-v 2294  df-sbc 2454

Copyright terms: Public domain