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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax-c7 | Structured version Visualization version GIF version |
Description: Axiom of Quantified
Negation. This axiom is used to manipulate negated
quantifiers. Equivalent to axiom scheme C7' in [Megill] p. 448 (p. 16 of
the preprint). An alternate axiomatization could use axc5c711 36696 in place
of ax-c5 36661, ax-c7 36663, and ax-11 2159.
This axiom is obsolete and should no longer be used. It is proved above as Theorem axc7 2317. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-c7 | ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . . . . 6 wff 𝜑 | |
2 | vx | . . . . . 6 setvar 𝑥 | |
3 | 1, 2 | wal 1541 | . . . . 5 wff ∀𝑥𝜑 |
4 | 3 | wn 3 | . . . 4 wff ¬ ∀𝑥𝜑 |
5 | 4, 2 | wal 1541 | . . 3 wff ∀𝑥 ¬ ∀𝑥𝜑 |
6 | 5 | wn 3 | . 2 wff ¬ ∀𝑥 ¬ ∀𝑥𝜑 |
7 | 6, 1 | wi 4 | 1 wff (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
This axiom is referenced by: ax10fromc7 36673 ax6fromc10 36674 equid1 36677 axc5c7 36689 axc711 36692 axc5c711 36696 equid1ALT 36703 |
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