Theorem axext4 2337

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 2334 and df-cleq 2346. (Contributed by NM, 14-Nov-2008.)

Assertion

Ref	Expression
axext4	⊢ (x = y ↔ ∀z(z ∈ x ↔ z ∈ y))

Distinct variable groups:   x,z   y,z

Proof of Theorem axext4

Step	Hyp	Ref	Expression
1		elequ2 1715	. . 3 ⊢ (x = y → (z ∈ x ↔ z ∈ y))
2	1	alrimiv 1631	. 2 ⊢ (x = y → ∀z(z ∈ x ↔ z ∈ y))
3		axext3 2336	. 2 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)
4	2, 3	impbii 180	1 ⊢ (x = y ↔ ∀z(z ∈ x ↔ z ∈ y))

Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ 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: (None)