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Theorem axext3 2633
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 2059, ax-12 2087, ax-13 2282. (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 2044 . . . . . 6 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
21bibi1d 332 . . . . 5 (𝑤 = 𝑥 → ((𝑧𝑤𝑧𝑦) ↔ (𝑧𝑥𝑧𝑦)))
32albidv 1889 . . . 4 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑤𝑧𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
4 ax-ext 2631 . . . 4 (∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦)
53, 4syl6bir 244 . . 3 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑤 = 𝑦))
6 ax7 1989 . . 3 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
75, 6syld 47 . 2 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
8 ax6ev 1947 . 2 𝑤 𝑤 = 𝑥
97, 8exlimiiv 1899 1 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  axext4  2635  dfcleq  2645  axextnd  9451  axextdist  31829  bj-cleqhyp  33017
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