Theorem axext3 2336

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 1715	. . . . 5 ⊢ (w = x → (z ∈ w ↔ z ∈ x))
2	1	bibi1d 310	. . . 4 ⊢ (w = x → ((z ∈ w ↔ z ∈ y) ↔ (z ∈ x ↔ z ∈ y)))
3	2	albidv 1625	. . 3 ⊢ (w = x → (∀z(z ∈ w ↔ z ∈ y) ↔ ∀z(z ∈ x ↔ z ∈ y)))
4		equequ1 1684	. . 3 ⊢ (w = x → (w = y ↔ x = y))
5	3, 4	imbi12d 311	. 2 ⊢ (w = x → ((∀z(z ∈ w ↔ z ∈ y) → w = y) ↔ (∀z(z ∈ x ↔ z ∈ y) → x = y)))
6		ax-ext 2334	. 2 ⊢ (∀z(z ∈ w ↔ z ∈ y) → w = y)
7	5, 6	chvarv 2013	1 ⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)

Syntax hints:   → wi 4   ↔ 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:  axext4  2337