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Theorem ax11o 1703
 Description: Derivation of set.mm's original ax-11o 1704 from the shorter ax-11 1397 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1695 or ax-17 1419. Normally, ax11o 1703 should be used rather than ax-11o 1704, 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 1397 . 2 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
21ax11a2 1702 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  ax11b  1707  equs5  1710
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