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Mirrors > Home > ILE Home > Th. List > ax11o | GIF version |
Description: Derivation of set.mm's
original ax-11o 1701 from the shorter ax-11 1394 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1692 or ax-17 1416. Normally, ax11o 1700 should be used rather than ax-11o 1701, 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 1394 | . 2 ⊢ (x = z → (∀zφ → ∀x(x = z → φ))) | |
2 | 1 | ax11a2 1699 | 1 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1240 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 |
This theorem is referenced by: ax11b 1704 equs5 1707 |
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