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Theorem axextg 2772
 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 2142, ax-12 2175, ax-13 2379. (Revised by BJ, 12-Jul-2019.) (Revised by Wolf Lammen, 9-Dec-2019.)
Assertion
Ref Expression
axextg (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axextg
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elequ2 2126 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
21bibi1d 347 . . . 4 (𝑤 = 𝑥 → ((𝑧𝑤𝑧𝑦) ↔ (𝑧𝑥𝑧𝑦)))
32albidv 1921 . . 3 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑤𝑧𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
4 equequ1 2032 . . 3 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
53, 4imbi12d 348 . 2 (𝑤 = 𝑥 → ((∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
6 ax-ext 2770 . 2 (∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦)
75, 6chvarvv 2005 1 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  axextb  2773  axextnd  10004  axextdist  33169
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