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Theorem axext4 2635
Description: A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2631 and df-cleq 2644. (Contributed by NM, 14-Nov-2008.)
Assertion
Ref Expression
axext4 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axext4
StepHypRef Expression
1 elequ2 2044 . . 3 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21alrimiv 1895 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
3 axext3 2633 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
42, 3impbii 199 1 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  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:  axc11next  38924
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