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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 38219 can be derived from ax-10 2136
(hbn1 2137), sp 2175 and propositional calculus. See ax10fromc7 38229 for the
rederivation of ax-10 2136 from ax-c7 38219.
Normally, axc7 2309 should be used rather than ax-c7 38219, 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 2175 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
2 | hbn1 2137 | . 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 1538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-ex 1781 |
This theorem is referenced by: modal-b 2311 axc10 2383 hbntg 35247 bj-modalb 36058 bj-axc10v 36135 axc5c4c711 43623 hbntal 43777 |
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