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Theorem axext3 2065
 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 1642 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
21bibi1d 231 . . . 4 (𝑤 = 𝑥 → ((𝑧𝑤𝑧𝑦) ↔ (𝑧𝑥𝑧𝑦)))
32albidv 1746 . . 3 (𝑤 = 𝑥 → (∀𝑧(𝑧𝑤𝑧𝑦) ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))
4 equequ1 1639 . . 3 (𝑤 = 𝑥 → (𝑤 = 𝑦𝑥 = 𝑦))
53, 4imbi12d 232 . 2 (𝑤 = 𝑥 → ((∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
6 ax-ext 2064 . 2 (∀𝑧(𝑧𝑤𝑧𝑦) → 𝑤 = 𝑦)
75, 6chvarv 1854 1 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1283 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-nf 1391 This theorem is referenced by:  axext4  2066
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