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Theorem axext3 2140
 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(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 2133 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
21bibi1d 232 . . . 4 (𝑤 = 𝑥 → ((𝑧𝑤𝑧𝑦) ↔ (𝑧𝑥𝑧𝑦)))
32albidv 1804 . . 3 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑤𝑧𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
4 equequ1 1692 . . 3 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
53, 4imbi12d 233 . 2 (𝑤 = 𝑥 → ((∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
6 ax-ext 2139 . 2 (∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦)
75, 6chvarv 1917 1 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-14 2131  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441 This theorem is referenced by:  axext4  2141
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