Theorem ssextss 2762

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 2755	. 2 |- (A (_ B <-> P~A (_ P~B)
2		dfss2 2058	. 2 |- (P~A (_ P~B <-> A.x(x e. P~A -> x e. P~B))
3		visset 1813	. . . . 5 |- x e. V
4	3	elpw 2404	. . . 4 |- (x e. P~A <-> x (_ A)
5	3	elpw 2404	. . . 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 999	. 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 954   e. wcel 958   (_ wss 2047  P~cpw 2401

This theorem is referenced by:  ssext 2763  nssss 2764

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412

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