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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 xz y) → x = y)
Distinct variable groups:   x,z   y,z

Proof of Theorem axext3
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 elequ2 1715 . . . . 5 (w = x → (z wz x))
21bibi1d 310 . . . 4 (w = x → ((z wz y) ↔ (z xz y)))
32albidv 1625 . . 3 (w = x → (z(z wz y) ↔ z(z xz y)))
4 equequ1 1684 . . 3 (w = x → (w = yx = y))
53, 4imbi12d 311 . 2 (w = x → ((z(z wz y) → w = y) ↔ (z(z xz y) → x = y)))
6 ax-ext 2334 . 2 (z(z wz y) → w = y)
75, 6chvarv 2013 1 (z(z xz y) → x = y)
 Colors of variables: wff setvar class 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
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