Theorem axext3 2336

Description: A generalization of the Axiom of Extensionality in which x and y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
axext3	|- (A.z(z e. x <-> z e. y) -> x = y)

Distinct variable groups:   x,z   y,z

Proof of Theorem axext3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 1715	. . . . 5 |- (w = x -> (z e. w <-> z e. x))
2	1	bibi1d 310	. . . 4 |- (w = x -> ((z e. w <-> z e. y) <-> (z e. x <-> z e. y)))
3	2	albidv 1625	. . 3 |- (w = x -> (A.z(z e. w <-> z e. y) <-> A.z(z e. x <-> z e. y)))
4		equequ1 1684	. . 3 |- (w = x -> (w = y <-> x = y))
5	3, 4	imbi12d 311	. 2 |- (w = x -> ((A.z(z e. w <-> z e. y) -> w = y) <-> (A.z(z e. x <-> z e. y) -> x = y)))
6		ax-ext 2334	. 2 |- (A.z(z e. w <-> z e. y) -> w = y)
7	5, 6	chvarv 2013	1 |- (A.z(z e. x <-> z e. y) -> x = y)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  axext4  2337