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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 38886 can be derived from ax-10 2141
(hbn1 2142), sp 2183 and propositional calculus. See ax10fromc7 38896 for the
rederivation of ax-10 2141 from ax-c7 38886.
Normally, axc7 2317 should be used rather than ax-c7 38886, 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 2183 | . 2 ⊢ (∀𝑥𝜑 → 𝜑) | |
| 2 | hbn1 2142 | . 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 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-ex 1780 |
| This theorem is referenced by: modal-b 2319 axc10 2390 hbntg 35806 bj-modalb 36717 bj-axc10v 36794 axc5c4c711 44420 hbntal 44573 |
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