Theorem ssextss 2768

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 2761	. 2 |- (A (_ B <-> P~A (_ P~B)
2		dfss2 2061	. 2 |- (P~A (_ P~B <-> A.x(x e. P~A -> x e. P~B))
3		visset 1816	. . . . 5 |- x e. V
4	3	elpw 2408	. . . 4 |- (x e. P~A <-> x (_ A)
5	3	elpw 2408	. . . 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 1001	. 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 956   e. wcel 960   (_ wss 2050  P~cpw 2405

This theorem is referenced by:  ssext 2769  nssss 2770

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416

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