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Theorem axextb 2710
Description: A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ax-ext 2707 and df-cleq 2728. (Contributed by NM, 14-Nov-2008.)
Assertion
Ref Expression
axextb (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem axextb
StepHypRef Expression
1 elequ2g 2123 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧𝑥𝑧𝑦))
2 axextg 2709 . 2 (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
31, 2impbii 209 1 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779
This theorem is referenced by:  dfcleq  2729  axc11next  44430
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