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Theorem axextndbi 33780
Description: axextnd 10347 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
Assertion
Ref Expression
axextndbi 𝑧(𝑥 = 𝑦 ↔ (𝑧𝑥𝑧𝑦))

Proof of Theorem axextndbi
StepHypRef Expression
1 axextnd 10347 . . 3 𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
2 elequ2 2121 . . . 4 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
32jctl 524 . . 3 (((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦) → ((𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦)) ∧ ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
41, 3eximii 1839 . 2 𝑧((𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦)) ∧ ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))
5 dfbi2 475 . . 3 ((𝑥 = 𝑦 ↔ (𝑧𝑥𝑧𝑦)) ↔ ((𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦)) ∧ ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
65exbii 1850 . 2 (∃𝑧(𝑥 = 𝑦 ↔ (𝑧𝑥𝑧𝑦)) ↔ ∃𝑧((𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦)) ∧ ((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)))
74, 6mpbir 230 1 𝑧(𝑥 = 𝑦 ↔ (𝑧𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wex 1782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-clel 2816  df-nfc 2889
This theorem is referenced by: (None)
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