Theorem axext3 1704

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

Step	Hyp	Ref	Expression
1		elequ2 1335	. . . . 5 |- (w = x -> (z e. w <-> z e. x))
2	1	bibi1d 678	. . . 4 |- (w = x -> ((z e. w <-> z e. y) <-> (z e. x <-> z e. y)))
3	2	albidv 1494	. . 3 |- (w = x -> (A.z(z e. w <-> z e. y) <-> A.z(z e. x <-> z e. y)))
4		equequ1 1332	. . 3 |- (w = x -> (w = y <-> x = y))
5	3, 4	imbi12d 685	. 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 1702	. 2 |- (A.z(z e. w <-> z e. y) -> w = y)
7	5, 6	chvarv 1550	1 |- (A.z(z e. x <-> z e. y) -> x = y)

Syntax hints:   -> wi 3   <-> wb 162  A.wal 1134   = wceq 1136   e. wcel 1138

This theorem is referenced by:  axext4 1706  axextnd 5891  axextdist 13656

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1143  ax-8 1144  ax-12 1148  ax-14 1150  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-ext 1702

This theorem depends on definitions:  df-bi 163  df-an 241

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