![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > axc7 | Structured version Visualization version GIF version |
Description: Show that the original
axiom ax-c7 35044 can be derived from ax-10 2135
(hbn1 2136) , sp 2167 and propositional calculus. See ax10fromc7 35054 for the
rederivation of ax-10 2135 from ax-c7 35044.
Normally, axc7 2292 should be used rather than ax-c7 35044, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.) |
Ref | Expression |
---|---|
axc7 | ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2167 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | hbn1 2136 | . 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | |
3 | 1, 2 | nsyl4 158 | 1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-12 2163 |
This theorem depends on definitions: df-bi 199 df-ex 1824 |
This theorem is referenced by: modal-b 2294 axc10 2349 hbntg 32303 bj-modalb 33298 bj-axc10v 33309 axc5c4c711 39567 hbntal 39723 |
Copyright terms: Public domain | W3C validator |