Theorem ssextss 2752

Description: An extensionality-like principle defining subclass in terms of subsets.

Assertion

Ref	Expression
ssextss	|- (A (_ B <-> A.x(x (_ A -> x (_ B))

Distinct variable groups:   x,A   x,B

Proof of Theorem ssextss

Step	Hyp	Ref	Expression
1		sspwb 2745	. 2 |- (A (_ B <-> P~A (_ P~B)
2		dfss2 2048	. 2 |- (P~A (_ P~B <-> A.x(x e. P~A -> x e. P~B))
3		visset 1804	. . . . 5 |- x e. V
4	3	elpw 2394	. . . 4 |- (x e. P~A <-> x (_ A)
5	3	elpw 2394	. . . 4 |- (x e. P~B <-> x (_ B)
6	4, 5	imbi12i 188	. . 3 |- ((x e. P~A -> x e. P~B) <-> (x (_ A -> x (_ B))
7	6	albii 996	. 2 |- (A.x(x e. P~A -> x e. P~B) <-> A.x(x (_ A -> x (_ B))
8	1, 2, 7	3bitr 177	1 |- (A (_ B <-> A.x(x (_ A -> x (_ B))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   e. wcel 955   (_ wss 2037  P~cpw 2391

This theorem is referenced by:  ssext 2753  nssss 2754

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402

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