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Mirrors > Home > MPE Home > Th. List > axc7 | Structured version Visualization version GIF version |
Description: Show that the original
axiom ax-c7 33188 can be derived from ax-10 2006
(hbn1 2007) , sp 2041 and propositional calculus. See ax10fromc7 33198 for the
rederivation of ax-10 2006 from ax-c7 33188.
Normally, axc7 2117 should be used rather than ax-c7 33188, 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 2041 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | hbn1 2007 | . 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | |
3 | 1, 2 | nsyl4 155 | 1 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: modal-b 2127 axc10 2240 hbntg 30955 bj-modalb 31893 bj-axc10v 31904 axc5c4c711 37624 hbntal 37790 |
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