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Theorem ax11o 1802
 Description: Derivation of set.mm's original ax-11o 1803 from the shorter ax-11 1486 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1794 or ax-17 1506. Normally, ax11o 1802 should be used rather than ax-11o 1803, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)
Assertion
Ref Expression
ax11o (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))

Proof of Theorem ax11o
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax-11 1486 . 2 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
21ax11a2 1801 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀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-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743 This theorem is referenced by:  ax11b  1806  equs5  1809
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