Theorem elabgtOLD 2401

Description: Membership in a class abstraction, using implicit substitition. (Closed theorem version of elabg 2405.)

Assertion

Ref	Expression
elabgtOLD	|- ((A e. B /\ A.x(x = A -> (ph <-> ps))) -> (A e. {x | ph} <-> ps))

Distinct variable groups:   x,A   ps,x

Proof of Theorem elabgtOLD

Step	Hyp	Ref	Expression
1		ax-17 1317	. . . . . 6 |- (y e. A -> A.x y e. A)
2		hbab1 1874	. . . . . 6 |- (y e. {x | ph} -> A.x y e. {x | ph})
3	1, 2	hbel 1996	. . . . 5 |- (A e. {x | ph} -> A.x A e. {x | ph})
4		ax-17 1317	. . . . 5 |- (ps -> A.xps)
5	3, 4	hbbi 1357	. . . 4 |- ((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
6	5	ax-gen 1305	. . 3 |- A.x((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
7		vtoclegft 2356	. . 3 |- ((A e. B /\ A.x((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps)) /\ A.x(x = A -> (A e. {x | ph} <-> ps))) -> (A e. {x | ph} <-> ps))
8	6, 7	mp3an2 1179	. 2 |- ((A e. B /\ A.x(x = A -> (A e. {x | ph} <-> ps))) -> (A e. {x | ph} <-> ps))
9		eleq1 1957	. . . . . . 7 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
10		abid 1873	. . . . . . 7 |- (x e. {x | ph} <-> ph)
11	9, 10	syl5rbbr 594	. . . . . 6 |- (x = A -> (A e. {x | ph} <-> ph))
12	11	bibi1d 681	. . . . 5 |- (x = A -> ((A e. {x | ph} <-> ps) <-> (ph <-> ps)))
13	12	biimprd 171	. . . 4 |- (x = A -> ((ph <-> ps) -> (A e. {x | ph} <-> ps)))
14	13	a2i 10	. . 3 |- ((x = A -> (ph <-> ps)) -> (x = A -> (A e. {x | ph} <-> ps)))
15	14	alimi 1338	. 2 |- (A.x(x = A -> (ph <-> ps)) -> A.x(x = A -> (A e. {x | ph} <-> ps)))
16	8, 15	sylan2 500	1 |- ((A e. B /\ A.x(x = A -> (ph <-> ps))) -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871

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-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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