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Theorem axext3 2603
 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 2016, ax-12 2044, ax-13 2245. (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 2001 . . . . . 6 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
21bibi1d 333 . . . . 5 (𝑤 = 𝑥 → ((𝑧𝑤𝑧𝑦) ↔ (𝑧𝑥𝑧𝑦)))
32albidv 1846 . . . 4 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑤𝑧𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
4 ax-ext 2601 . . . 4 (∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦)
53, 4syl6bir 244 . . 3 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑤 = 𝑦))
6 ax7 1940 . . 3 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
75, 6syld 47 . 2 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
8 ax6ev 1887 . 2 𝑤 𝑤 = 𝑥
97, 8exlimiiv 1856 1 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by:  axext4  2605  dfcleq  2615  axextnd  9373  axextdist  31459  bj-cleqhyp  32592
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