Theorem ssintab 2550

Description: Subclass of the intersection of a class abstraction.

Assertion

Ref	Expression
ssintab	|- (A (_ |^|{x | ph} <-> A.x(ph -> A (_ x))

Distinct variable group:   x,A

Proof of Theorem ssintab

Step	Hyp	Ref	Expression
1		ssint 2549	. 2 |- (A (_ |^|{x | ph} <-> A.y e. {x | ph}A (_ y)
2		df-ral 1649	. 2 |- (A.y e. {x | ph}A (_ y <-> A.y(y e. {x | ph} -> A (_ y))
3		hbab1 1466	. . . . 5 |- (y e. {x | ph} -> A.x y e. {x | ph})
4		ax-17 971	. . . . 5 |- (A (_ y -> A.x A (_ y)
5	3, 4	hbim 1007	. . . 4 |- ((y e. {x | ph} -> A (_ y) -> A.x(y e. {x | ph} -> A (_ y))
6		ax-17 971	. . . 4 |- ((x e. {x | ph} -> A (_ x) -> A.y(x e. {x | ph} -> A (_ x))
7		eleq1 1534	. . . . 5 |- (y = x -> (y e. {x | ph} <-> x e. {x | ph}))
8		sseq2 2083	. . . . 5 |- (y = x -> (A (_ y <-> A (_ x))
9	7, 8	imbi12d 626	. . . 4 |- (y = x -> ((y e. {x | ph} -> A (_ y) <-> (x e. {x | ph} -> A (_ x)))
10	5, 6, 9	cbval 1165	. . 3 |- (A.y(y e. {x | ph} -> A (_ y) <-> A.x(x e. {x | ph} -> A (_ x))
11		abid 1465	. . . . 5 |- (x e. {x | ph} <-> ph)
12	11	imbi1i 186	. . . 4 |- ((x e. {x | ph} -> A (_ x) <-> (ph -> A (_ x))
13	12	albii 999	. . 3 |- (A.x(x e. {x | ph} -> A (_ x) <-> A.x(ph -> A (_ x))
14	10, 13	bitr 173	. 2 |- (A.y(y e. {x | ph} -> A (_ y) <-> A.x(ph -> A (_ x))
15	1, 2, 14	3bitr 177	1 |- (A (_ |^|{x | ph} <-> A.x(ph -> A (_ x))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645   (_ wss 2047  |^|cint 2533

This theorem is referenced by:  ssmin 2552  intmin4 2559

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-10 966  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-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-int 2534

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