Theorem axext4 2021

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 2019. (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 1598	. . 3 ⊢ (x = y → (z ∈ x ↔ z ∈ y))
2	1	alrimiv 1751	. 2 ⊢ (x = y → ∀z(z ∈ x ↔ z ∈ y))
3		axext3 2020	. 2 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)
4	2, 3	impbii 117	1 ⊢ (x = y ↔ ∀z(z ∈ x ↔ z ∈ y))

Syntax hints:   ↔ wb 98  ∀wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by: (None)