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Mirrors > Home > MPE Home > Th. List > hba1 | Structured version Visualization version GIF version |
Description: The setvar 𝑥 is not free in ∀𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Oct-2021.) |
Ref | Expression |
---|---|
hba1 | ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2152 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
2 | 1 | nf5ri 2193 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-or 845 df-ex 1782 df-nf 1786 |
This theorem is referenced by: axi5r 2762 axial 2763 bj-19.41al 34105 bj-wnf1 34164 hbntal 41259 hbimpg 41260 hbimpgVD 41610 hbalgVD 41611 hbexgVD 41612 ax6e2eqVD 41613 e2ebindVD 41618 vk15.4jVD 41620 |
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