Theorem axext3 2009

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	⊢ (∀z(z ∈ x ↔ z ∈ 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 1589	. . . . 5 ⊢ (w = x → (z ∈ w ↔ z ∈ x))
2	1	bibi1d 222	. . . 4 ⊢ (w = x → ((z ∈ w ↔ z ∈ y) ↔ (z ∈ x ↔ z ∈ y)))
3	2	albidv 1693	. . 3 ⊢ (w = x → (∀z(z ∈ w ↔ z ∈ y) ↔ ∀z(z ∈ x ↔ z ∈ y)))
4		equequ1 1586	. . 3 ⊢ (w = x → (w = y ↔ x = y))
5	3, 4	imbi12d 223	. 2 ⊢ (w = x → ((∀z(z ∈ w ↔ z ∈ y) → w = y) ↔ (∀z(z ∈ x ↔ z ∈ y) → x = y)))
6		ax-ext 2008	. 2 ⊢ (∀z(z ∈ w ↔ z ∈ y) → w = y)
7	5, 6	chvarv 1798	1 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1231

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 1321  ax-gen 1323  ax-ie1 1369  ax-ie2 1370  ax-8 1382  ax-4 1387  ax-14 1392  ax-17 1406  ax-i9 1410  ax-ial 1415  ax-ext 2008

This theorem depends on definitions:  df-bi 110  df-nf 1335

This theorem is referenced by:  axext4  2010