Theorem axext4 1869

Description: A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 1865 and df-cleq 1877.

Assertion

Ref	Expression
axext4	|- (x = y <-> A.z(z e. x <-> z e. y))

Distinct variable groups:   x,z   y,z

Proof of Theorem axext4

Step	Hyp	Ref	Expression
1		elequ2 1497	. . 3 |- (x = y -> (z e. x <-> z e. y))
2	1	19.21aiv 1664	. 2 |- (x = y -> A.z(z e. x <-> z e. y))
3		axext3 1867	. 2 |- (A.z(z e. x <-> z e. y) -> x = y)
4	2, 3	impbii 174	1 |- (x = y <-> A.z(z e. x <-> z e. y))

Syntax hints:   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300

This theorem is referenced by:  ax10ext 16364

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-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

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