Theorem elabgt 1867

Description: Membership in a class abstraction with implicit substitution. (Closed theorem version of elabg 1871.)

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

Proof of Theorem elabgt

Step	Hyp	Ref	Expression
1		ax-17 1190	. . . . . 6 |- (y e. A -> A.x y e. A)
2		hbab1 1443	. . . . . 6 |- (y e. {x | ph} -> A.x y e. {x | ph})
3	1, 2	hbel 1542	. . . . 5 |- (A e. {x | ph} -> A.x A e. {x | ph})
4		ax-17 1190	. . . . 5 |- (ps -> A.xps)
5	3, 4	hbbi 986	. . . 4 |- ((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
6	5	ax-gen 955	. . 3 |- A.x((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
7		vtoclegft 1831	. . 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 900	. 2 |- ((A e. B /\ A.x(x = A -> (A e. {x | ph} <-> ps))) -> (A e. {x | ph} <-> ps))
9		eleq1 1510	. . . . . . 7 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
10		abid 1442	. . . . . . 7 |- (x e. {x | ph} <-> ph)
11	9, 10	syl5rbbr 533	. . . . . 6 |- (x = A -> (A e. {x | ph} <-> ph))
12	11	bibi1d 617	. . . . 5 |- (x = A -> ((A e. {x | ph} <-> ps) <-> (ph <-> ps)))
13	12	biimprd 154	. . . 4 |- (x = A -> ((ph <-> ps) -> (A e. {x | ph} <-> ps)))
14	13	a2i 9	. . 3 |- ((x = A -> (ph <-> ps)) -> (x = A -> (A e. {x | ph} <-> ps)))
15	14	19.20i 968	. 2 |- (A.x(x = A -> (ph <-> ps)) -> A.x(x = A -> (A e. {x | ph} <-> ps)))
16	8, 15	sylan2 451	1 |- ((A e. B /\ A.x(x = A -> (ph <-> ps))) -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950   = wceq 1099   e. wcel 1105  {cab 1440

This theorem is referenced by:  sbcel12g 1982  sbceqdig 1983

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787

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