Theorem elabgt 2982

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2986.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

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

Distinct variable groups:   x,A   ps,x

Allowed substitution hints:   ph(x)   B(x)

Proof of Theorem elabgt

Step	Hyp	Ref	Expression
1		abid 2341	. . . . . . 7 |- (x e. {x | ph} <-> ph)
2		eleq1 2413	. . . . . . 7 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
3	1, 2	syl5bbr 250	. . . . . 6 |- (x = A -> (ph <-> A e. {x | ph}))
4	3	bibi1d 310	. . . . 5 |- (x = A -> ((ph <-> ps) <-> (A e. {x | ph} <-> ps)))
5	4	biimpd 198	. . . 4 |- (x = A -> ((ph <-> ps) -> (A e. {x | ph} <-> ps)))
6	5	a2i 12	. . 3 |- ((x = A -> (ph <-> ps)) -> (x = A -> (A e. {x | ph} <-> ps)))
7	6	alimi 1559	. 2 |- (A.x(x = A -> (ph <-> ps)) -> A.x(x = A -> (A e. {x | ph} <-> ps)))
8		nfcv 2489	. . . 4 |- F/_xA
9		nfab1 2491	. . . . . 6 |- F/_x{x | ph}
10	9	nfel2 2501	. . . . 5 |- F/x A e. {x | ph}
11		nfv 1619	. . . . 5 |- F/xps
12	10, 11	nfbi 1834	. . . 4 |- F/x(A e. {x | ph} <-> ps)
13		pm5.5 326	. . . 4 |- (x = A -> ((x = A -> (A e. {x | ph} <-> ps)) <-> (A e. {x | ph} <-> ps)))
14	8, 12, 13	spcgf 2934	. . 3 |- (A e. B -> (A.x(x = A -> (A e. {x | ph} <-> ps)) -> (A e. {x | ph} <-> ps)))
15	14	imp 418	. 2 |- ((A e. B /\ A.x(x = A -> (A e. {x | ph} <-> ps))) -> (A e. {x | ph} <-> ps))
16	7, 15	sylan2 460	1 |- ((A e. B /\ A.x(x = A -> (ph <-> ps))) -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by: (None)