Theorem axext3OLD 1868

Description: A generalization of the Axiom of Extensionality in which x and y need not be distinct.

Assertion

Ref	Expression
axext3OLD	|- (A.z(z e. x <-> z e. y) -> x = y)

Distinct variable groups:   x,z   y,z

Proof of Theorem axext3OLD

Step	Hyp	Ref	Expression
1		a9e 1483	. 2 |- E.w w = x
2		ax-ext 1865	. . . 4 |- (A.z(z e. w <-> z e. y) -> w = y)
3		elequ2 1497	. . . . . . 7 |- (w = x -> (z e. w <-> z e. x))
4	3	bibi1d 681	. . . . . 6 |- (w = x -> ((z e. w <-> z e. y) <-> (z e. x <-> z e. y)))
5	4	albidv 1656	. . . . 5 |- (w = x -> (A.z(z e. w <-> z e. y) <-> A.z(z e. x <-> z e. y)))
6		equequ1 1494	. . . . 5 |- (w = x -> (w = y <-> x = y))
7	5, 6	imbi12d 688	. . . 4 |- (w = x -> ((A.z(z e. w <-> z e. y) -> w = y) <-> (A.z(z e. x <-> z e. y) -> x = y)))
8	2, 7	mpbii 210	. . 3 |- (w = x -> (A.z(z e. x <-> z e. y) -> x = y))
9	8	19.23aiv 1674	. 2 |- (E.w w = x -> (A.z(z e. x <-> z e. y) -> x = y))
10	1, 9	ax-mp 7	1 |- (A.z(z e. x <-> z e. y) -> x = y)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-8 1306  ax-9 1307  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

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