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Mirrors > Home > ILE Home > Th. List > ax11o | GIF version |
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.) |
Ref | Expression |
---|---|
ax11o | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-11 1397 | . 2 ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
2 | 1 | ax11a2 1702 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
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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