Theorem elrabsf 1963

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 1904 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 groups:   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-17 971	. . . 4 |- (y e. B -> A.z y e. B)
3		ax-17 971	. . . 4 |- (ph -> A.zph)
4		hbs1 1332	. . . 4 |- ([z / x]ph -> A.x[z / x]ph)
5		sbequ12 1181	. . . 4 |- (x = z -> (ph <-> [z / x]ph))
6	1, 2, 3, 4, 5	cbvrab 1910	. . 3 |- {x e. B | ph} = {z e. B | [z / x]ph}
7	6	eleq2i 1538	. 2 |- (A e. {x e. B | ph} <-> A e. {z e. B | [z / x]ph})
8		ax-17 971	. . . 4 |- (w e. A -> A.z w e. A)
9		ax-17 971	. . . 4 |- (w e. B -> A.z w e. B)
10	8	hbsbc1 1949	. . . 4 |- ((A e. V -> [A / z][z / x]ph) -> A.z(A e. V -> [A / z][z / x]ph))
11		sbceq1a 1944	. . . . 5 |- (z = A -> ([z / x]ph <-> [A / z][z / x]ph))
12		19.8a 1029	. . . . . . 7 |- (z = A -> E.z z = A)
13		isset 1814	. . . . . . 7 |- (A e. V <-> E.z z = A)
14	12, 13	sylibr 200	. . . . . 6 |- (z = A -> A e. V)
15		biimt 731	. . . . . 6 |- (A e. V -> ([A / z][z / x]ph <-> (A e. V -> [A / z][z / x]ph)))
16	14, 15	syl 10	. . . . 5 |- (z = A -> ([A / z][z / x]ph <-> (A e. V -> [A / z][z / x]ph)))
17	11, 16	bitrd 528	. . . 4 |- (z = A -> ([z / x]ph <-> (A e. V -> [A / z][z / x]ph)))
18	8, 9, 10, 17	elrabf 1904	. . 3 |- (A e. {z e. B | [z / x]ph} <-> (A e. B /\ (A e. V -> [A / z][z / x]ph)))
19		elisset 1817	. . . . 5 |- (A e. B -> A e. V)
20	19, 15	syl 10	. . . 4 |- (A e. B -> ([A / z][z / x]ph <-> (A e. V -> [A / z][z / x]ph)))
21	20	pm5.32i 645	. . 3 |- ((A e. B /\ [A / z][z / x]ph) <-> (A e. B /\ (A e. V -> [A / z][z / x]ph)))
22	18, 21	bitr4 176	. 2 |- (A e. {z e. B | [z / x]ph} <-> (A e. B /\ [A / z][z / x]ph))
23		sbccog 1952	. . 3 |- (A e. B -> ([A / z][z / x]ph <-> [A / x]ph))
24	23	pm5.32i 645	. 2 |- ((A e. B /\ [A / z][z / x]ph) <-> (A e. B /\ [A / x]ph))
25	7, 22, 24	3bitr 177	1 |- (A e. {x e. B | ph} <-> (A e. B /\ [A / x]ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  {crab 1648  Vcvv 1811

This theorem is referenced by:  elabs2 1964  iunrab 2596  reucl2 2888  onminesb 3010  tfis 3127

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rab 1652  df-v 1812  df-sbc 1942

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