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Theorem axext3 2592
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.) Remove dependencies on ax-10 2006, ax-12 2034, ax-13 2234. (Revised by Wolf Lammen, 9-Dec-2019.)
Assertion
Ref Expression
axext3 (∀𝑧(𝑧𝑥𝑧𝑦) → 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 1991 . . . . . 6 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
21bibi1d 332 . . . . 5 (𝑤 = 𝑥 → ((𝑧𝑤𝑧𝑦) ↔ (𝑧𝑥𝑧𝑦)))
32albidv 1836 . . . 4 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑤𝑧𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
4 ax-ext 2590 . . . 4 (∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦)
53, 4syl6bir 243 . . 3 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑤 = 𝑦))
6 ax7 1930 . . 3 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
75, 6syld 46 . 2 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
8 ax6ev 1877 . 2 𝑤 𝑤 = 𝑥
97, 8exlimiiv 1846 1 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696
This theorem is referenced by:  axext4  2594  axextnd  9292  axextdist  30949  bj-cleqhyp  32084
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