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Theorem ax11o 1836
Description: Derivation of set.mm's original ax-11o 1837 from the shorter ax-11 1520 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1828 or ax-17 1540.

Normally, ax11o 1836 should be used rather than ax-11o 1837, 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 1520 . 2 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
21ax11a2 1835 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548
This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777
This theorem is referenced by:  ax11b  1840  equs5  1843
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